Paper Title: Dimension formulas for modular form spaces of rational weights, the classification of eta-quotient characters and an extension of Martin's theorem.





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Dimension formulas for modular form spaces]Dimension formulas for modular form spaces of rational weights, the classification of eta-quotient characters and an extension of Martin's theorem


School of Mathematical Sciences

Key Laboratory of MEA (Ministry of Education) & Shanghai Key Laboratory of PMMP

East China Normal University

500 Dongchuan Road, 200241

Shanghai, P. R. China
zhuxiaojiemath@outlook.com
https://orcid.org/0000-0002-6733-0755




§ ABSTRACT 
We give an explicit formula for dimensions of spaces of rational-weight modular forms whose multiplier systems are induced by eta-quotients of fractional exponents. As the first application, we give series expressions of Fourier coefficients of the n-th root of certain infinite q-products. As the second application, we extend Yves Martin's list of multiplicative holomorphic eta-quotients of integral weights by first extending the meaning of multiplicativity, then identifying one-dimensional spaces, and finally applying Wohlfahrt's extension of Hecke operators. A table containing 2277 of such eta-quotients is presented. As a related result, we completely classify the multiplier systems induced by eta-quotients of integral exponents. For instance, there are totally 384 such multiplier systems on Γ_0(4) for any fixed weight. We also provide SageMath programs on checking the theorems and generating the tables.


[2020]Primary 11F12; Secondary 11F20, 11F25, 11F30, 11L05, 30F10



This work is supported in part by Science and Technology Commission of Shanghai Municipality (No. 22DZ2229014).

[
    Xiao-Jie Zhu
    
================








§ INTRODUCTION




 §.§ Dimension formulas

A holomorphic function f defined on the upper half plane  is called a modular form if it satisfies two conditions: first f(aτ+b/cτ+d)=(cτ+d)^kχabcdf(τ), τ∈ for all abcd in some discrete matrix group G, k being an integer or half integer, called the weight, and χ being a multiplier system of G; second f is holomorphic at all cusps of G, the meaning of which will be clarified in Definition <ref>. Let the space of all modular forms with given G, k and χ be denoted by M_k(G,χ).

Modular forms, spaces of modular forms and their variants play important roles in mathematics. In the study of these topics, having an explicit formula for the dimension of the complex vector space M_k(G,χ) is crucial to some applications. For instance, _ M_k(G,χ)=1 would imply that any function f in M_k(G,χ) is an eigenfunction for a family of operators called the Hecke operators and hence we obtain nontrivial relations among the Fourier coefficients of f.

A natural and elegant way for computing _ M_k(G,χ) is to use the Riemann-Roch theorem and the Riemann-Hurwitz formula. Petersson <cit.> applied this method and gave a dimension formula which he called the generalized Riemann-Roch theorem. Petersson's formula concerns arbitrary Fuchsian group G of the first kind, arbitrary complex weight k and arbitrary multiplier system χ while this formula is not so explicit in the sense one can not directly compute the dimension using elementary operations and numerical information about G and χ. To our best knowledge, Shimura <cit.> gave explicit formulas for G arbitrary, k being an integer and χ trivial first. Another way for computing _ M_k(G,χ) is to use the Eichler-Selberg trace formula for Hecke operators; c.f. <cit.> for the case G being the full modular group, χ trivial and <cit.> for the case G=Γ_0(N) and χ induced by a Dirichlet character modulo N, both dealing with integral weights. See also <cit.>. For the half-integral weights, Cohen and Oesterlé <cit.> gave an explicit dimension formula for the case G=Γ_0(4N), χ being the product of the multiplier system of the theta series ∑_n∈^2 n^2τ and that of a Dirichlet character modulo 4N. Since then, there appear many excellent works on computing _ M_k(G,χ) for certain special G or special χ, or computing the dimensions of certain subspaces; c.f. <cit.>, <cit.>, <cit.> and <cit.>.

Modular forms of rational weights are less attractive to mathematicians than those of integral or half-integral weights. We know little about them. For instance, it seems no Hecke theory has been attempted. (Maybe there is no such theory.) However, there actually exist many such forms and more importantly, there are interesting applications, e.g., to noncongruence modular forms. Explicit dimension formulas for spaces of rational weights have been obtained by Ibukiyama <cit.>, <cit.> in some special cases, using essentially the method of Petersson <cit.>, but no proof is given.

As our main result, we give explicit formulas for _ M_k(Γ_0(N),χ) where k is any rational number, χ is the multiplier system of any level N eta-quotient ∏_n| Nη(nτ)^r_n of fractional exponents r_n. See Definition <ref> for the exact meaning of M_k(Γ_0(N),χ) and (<ref>) for that of χ. The proof is based on Petersson's method <cit.> and we provide the full details. The following theorem is a special case of Theorem <ref>, which is our main theorem.

If k>2-6/mε_2-8/mε_3-12/m∑_c| Nϕ(c,N/c)·(1-{x_c/24}), then

    _ M_k(Γ_0(N), χ)=k-1/12m+1/4ε_2+1/3ε_3+∑_c| Nϕ(c,N/c)·(1/2-{x_c/24}),

where m, ε_2 and ε_3, depending on N, are given in (<ref>), (<ref>) and (<ref>) respectively and

    k=1/2∑_n| Nr_n,   x_c=∑_n| NN/(N,c^2)·(n,c)^2/nr_n    for c| N.

Note that the notation ϕ(c,N/c) refers to the Euler totient ϕ of the greatest common divisor of c and N/c.


In particular, if k≥2, then (<ref>) always holds. For any rational k with 0<k<2, there are also infinitely many pairs (N,χ) such that _ M_k(Γ_0(N), χ) can be computed using (<ref>). See Section <ref> for complete lists of such pairs (N,χ) for k=1/2, 1, 3/2 and r_n being integers. The dimensions for k=1/2 and k=3/2 in the case G=Γ_0(4N) and χ is the product of the multiplier system of the theta series ∑_n∈^2 n^2τ and that of a Dirichlet character modulo 4N are previously known: Serre and Starks <cit.> gave explicit bases of the spaces of weight 1/2; Cohen and Oesterlé <cit.> established relations between spaces of weight 1/2 and those of weight 3/2. The important and interesting problem of computing dimensions in weight 1 is difficult and the most complete result about this was obtained by Deligne and Serre <cit.>. Using their theory, one can compute _ M_1(Γ_0(N), χ) for any N∈Z1 and χ induced by any Dirichlet character modulo N although there is no explicit formula.

The difference between our formulas for _ M_k(Γ_0(N), χ) with k∈1/2 and others' is that we deal with those χ induced by eta-quotients while the existing formulas are about those χ induced by Dirichlet characters or the multiplier system of ∑_n∈^2 n^2τ. Moreover, when k∈∖1/2, our formulas include the one in <cit.> and <cit.> as special cases.

After preparing, stating and proving the main theorem and establishing some related tools (Proposition <ref>, Theorem <ref>, Tables <ref>, <ref> and Section <ref>), we will give mainly two applications, both of which are based on identifying one-dimensional spaces. The first application concerns rational-weight modular forms and the second concerns ordinary eta-quotients whose exponents r_n are all integers.



 §.§ Application I


We give a series expression of the Fourier coefficients of the n-th root of certain infinite q-products. See Corollary <ref> for the details. To avoid overlapping with the main text, here we give a randomly chosen example which is different from Examples <ref> and <ref>.


We have

    √(∏_n∈Z1(1-q^n))·√(∏_n∈Z1(1-q^2n)^29)/√(∏_n∈Z1(1-q^4n)^22)=1+^-15/8(2)^15/4/Γ(15/4)
    ·∑_n∈Z1n^11/4(∑_c∈Z11/(4c)^15/4∑_0≤ d<4c(d,4c)=1dn/4c-1/24P(4c,d;29/2,-22/3))· q^n,

where q=^2τ throughout the paper, τ∈ and

    P(4c,d;29/2,-22/3)=-88(-d,c)+174(-d,2c)+4s(-d,4c)-45/2,
    
    s(-d,c)=∑_r=1^c-1r/c({-dr/c}-1/2).



Corollary <ref> contains infinitely many such identities of level 4, extending a previous result of the author <cit.> which concerns levels 2 and 3. For the outline of the proof, say of Proposition <ref>, let the infinite product be f, which is an eta-quotient of fractional exponents. Then f∈ M_15/4(Γ_0(4),χ) where χ is the multiplier system of f. There is as well an Eisenstein series g in M_15/4(Γ_0(4),χ); c.f. Definition <ref>. Now we apply Theorem <ref> and find that _ M_15/4(Γ_0(4),χ)=1. Hence f and g are proportional. The identity thus follows by figuring out the Fourier coefficients of g.




 §.§ Application II


This is the major application. In 1996, Martin <cit.> obtained the complete list[There is a misprint in <cit.>. The entry 2^-1·4^4·6^-1·8^-1·12^4·24 should be corrected to 2^-1·4^4·6^-1·8^-1·12^4·24^-1.] of integral-weight holomorphic eta-quotients f (with integral exponents) satisfying that

  
  * the multiplier system of f is induced by some Dirichlet character,
  
  * if we write f=∑_n∈Z0c_f(n)q^n, then c_f(n_1)c_f(n_2)=c_f(n_1n_2)c_f(1) for any coprime positive integers.

We will discard condition (a) and seek for arbitrary holomorphic eta-quotients (with integral exponents) satisfying a multiplicativity property similar to the one in condition (b). We will also give many interesting new identities involving Fourier coefficients of these eta-quotients. Let us first present some randomly chosen examples. For the basic knowledge about Dedekind eta function η(τ) and eta-quotients, see the second half of Section <ref>.

Let f(τ)=η(τ)^-7η(2τ)^17η(4τ)^-3=∑_n∈5/8+Z0c_f(n)q^n. Let l_1,l_2 be coprime odd square-free integers. Then

    c_f(5l_1^2/8)c_f(5l_2^2/8)=c_f(5l_1^2l_2^2/8).

See Proposition <ref> for details and the proof.



Let f(τ)=η(3τ)^2η(9τ)^-1η(27τ)=∑_n∈Z1c_f(n)q^n. Then c_f(l_1)c_f(l_2)=c_f(l_1l_2) whenever l_1,l_2≡13 are square-free positive integers with (l_1,l_2)=1. See Example <ref> for details.



Let f(τ)=η(τ)^1η(2τ)^-1η(3τ)^-1η(4τ)^1η(6τ)^4η(12τ)^-2=∑_n∈Z0c_f(n)q^n. Then

    c_f(l)=-∑_a| la6=-∏_p| l(1+p6)

where 1≤ l≡124 is square-free. Note that a6 refers to the Kronecker-Jacobi symbol. See Example <ref> for details.


Martin <cit.> proved his results using Hecke operators in the sense of <cit.> while we prove our results using Wohlfahrt's extension of Hecke operators <cit.>. See also <cit.> for a theory of the double coset version of these operators. These operators are denoted by T_l, where l runs through a multiplicative submonoid L of the positive integers; c.f. Corollary <ref>. As a prerequisite, we give an explicit formula for the action on Fourier coefficients; c.f. Theorem <ref>. Then we aim to find holomorphic eta-quotient f such that T_lf=c_l· f with c_l∈. At this point Theorem <ref> enters into play. If f, whose level is N and character is χ, lies in a one-dimensional space M_k(Γ_0(N),χ) or S_k(Γ_0(N),χ), the subspace of cusp forms, then we will have T_lf=c_l· f for all l∈ L. (L depends on f.) This fact, which is stated in Theorem <ref>, is the main theorem of this application.

It seems that there are more than 10000 eta-quotients that are Hecke eigenforms (in the sense of (<ref>)) for infinitely many l. We list 2277 of them in Table <ref>. They are what we call admissible eta-quotients of type I with levels

    N=1,2,3,4,5,6,7,8,9,10,11,13,14,15,17,19,21,27.

For the meaning of admissible eta-quotients of type I, see the beginning part of Section <ref>. For other levels, either there is no admissible eta-quotient of type I, or there are too many so it is not appropriate to list them in the paper. One can find the SageMath code that generates admissible eta-quotients of type I or II with any given level in Appendix <ref>.

Besides Theorem <ref>, another important result is Theorem <ref>, which says that

    c_l_1· c_l_2=c_l_1l_2,   l_1,l_2∈ L,  (l_1,l_2)=1.

Here c_l=T_lf/f. This is the genuine multiplicativity property that is satisfied by all admissible eta-quotients and reduces to the relation c_f(n_1)c_f(n_2)=c_f(n_1n_2)c_f(1) for ordinary multiplicative eta-quotients in <cit.>.

All identities in the above three examples are immediate consequences of Theorems <ref> and <ref>.



 §.§ Other results, structure of the paper and notations

As a related result, we give the complete classification of linear characters that are induced by eta-quotients ∏_n| Nη(nτ)^r_n with r_n∈ on the double cover of Γ_0(N). See Corollary <ref> for the conclusion and Examples <ref>, <ref>, <ref>, <ref>, <ref> for examples. For instance, for each k∈1/2, there are totally 384 linear characters of the double cover of Γ_0(4) that are induced by η(τ)^r_1η(2τ)^r_2η(4τ)^r_n of weight k. As a comparison, there are only two characters of Γ_0(4) that are induced by Dirichlet characters modulo 4 and these two characters are also contained in the above 384 characters with any fixed k∈.

The structure of the paper is as follows. We review some elements of rational-weight modular forms in Section <ref>. The second half of this section contains elements of Dedekind eta function and eta-quotients of fractional exponents. In Section <ref> we associate a divisor on certain compact Riemann surface with any rational-weight meromorphic modular form and in addition, we give a detailed proof of the valence formula in the case of rational weights. In Section <ref> we state and prove the main theorem via Petersson's method <cit.>. The following three sections contain some tools which are needed by the following two applications: Section <ref> contains a formula that relates the orders at cusps and the exponents (or the character) of an eta-quotient. This formula is due to Bhattacharya (cf. <cit.>). Here we give more details on the proof. In Section <ref> we classify all linear characters that are induced by eta-quotients ∏_n| Nη(nτ)^r_n with r_n∈ on the double cover of Γ_0(N), as is described in the last paragraph. Section <ref>, which is subdivided into three subsections, contains tables of dimensions of modular form spaces of weight 1/2, 1 and 3/2 that can be computed by our main theorem. In Sections <ref> and <ref> we carry out the first and second applications, which has been described in Sections <ref> and <ref>, respectively. Section <ref> contains some comments, open problems and conjectures. Finally, many formulas in this paper have been verified and many tables are generated by SageMath <cit.> programs. Appendix <ref> contains the usage of the code.

We collect some notations. For a set A, the symbols A and #A both denote its cardinality. The notation ^A denotes the -vector space of functions from A into . A multiset {x_i i∈ I} is defined to be the ordinary set {(y,n) n=#{i∈ I x_i=y}, n≠0}. Its underlying (ordinary) set is still denoted by {x_i i∈ I}. If f X→ Y is a function, then f|_A is its restriction to A where A is a subset of X. If f,g are functions then f∘ g is their composition: f∘ g(x)=f(g(x)). For a family of sets A_i, i∈ I, the notation _i∈ IA_i denotes the union and only when the family A_i are disjoint can one use this notation.

For a group G, ⟨ X⟩ denotes the subgroup generated by X where X is a subset of G. If H is a subgroup of G then [G H]=#G/H=#H\ G is the index. If G acts on the left (right resp.) on a set X, then G\ X (X/G resp.) denotes the set of orbits G· x (x· G resp.) where x∈ X. If X is itself a group, G is a subgroup of X and the action is the group operation, then G\ X (X/G resp.) is called the left coset space (right coset space resp.) For x∈ X, G_x denotes the stabilizer, that is, the subgroup of g∈ G such that gx=x (or xg=x). A character χ of G is a complex linear character, that is, a group homomorphism from G to ^×, the multiplicative group of nonzero complex numbers. If χ(g)=1 for all g∈ G, then χ is called unitary. If G' is another group and π G→ G' is a group surjection, then we say the character χ descends to a character on G' if there is a character χ' of G' such that χ=χ'∘π.

The symbol ϕ refers to the Euler totient function. For integers a, b,…, the notation (a,b,…) denotes the greatest common divisor and ϕ(a,b,…) denotes the value of ϕ at (a,b,…). Let N∈Z1; the summation range of ∑_n| N is implicitly understood to be the positive divisors of N. Let p be a prime and α∈Z0; then p^α∥ N means p^α| N but p^α+1∤ N. The p-adic exponential valuation, v_p(N), is the largest α such that p^α∥ N. An empty sum is understood to be 0 and an empty product be 1. For a real number x, the notation [x] means the largest integer not exceeding x and {x}=x-[x]. The functions Γ(s) and ζ(s) are the usual Euler Gamma function and Riemann zeta function respectively.

Let m,n∈; the Kronecker-Jacobi symbol mn is defined as follows:


  * mp is the usual Legendre symbol if p is an odd prime.

  * m2 equals 0 if 2 | m, and equals (-1)^(m^2-1)/8 if 2 ∤ m.

  * m-1 equals 1 if m ≥ 0, and equals -1 otherwise.

  * m1=1 by convention.

  * mn is defined to make it a complete multiplicative function of n ∈-{0}.

  * m0=0 if m ≠±1, and ±10=1.

We shall freely use the following properties, especially in the proof of Theorem <ref>.


  * mn_1n_2=mn_1mn_2. When m=-1, it is required that n_1,n_2≠0.

  * m_1m_2n=m_1nm_2n. When n=-1, it is required that either m_1,m_2≠0, or one of m_1, m_2 is 0 and the other is nonnegative.

  * mnnm=ε(m,n)·(-1)^n-1/2·m-1/2 where m,n are coprime odd integers, ε(m,n)=-1 if n,m<0 and ε(m,n)=1 otherwise.

  * -1n=(-1)^n-1/2 and 2n=(-1)^n^2-1/8 where n is odd.

  * The function n↦mn is m-periodic if m≡0,14; it is 4m-periodic if m≡24.

  * The function m↦mn is n-periodic if n is odd and positive.

For the proofs, see <cit.>.




§ MODULAR FORMS OF RATIONAL WEIGHT


In this section, we review some elements of the theory of modular forms of rational weight (cf. <cit.>). The concept of modular forms of rational weight is a special case of that of generalized modular forms which was initiated by Knopp and Mason <cit.>. The multiplier system of a generalized modular form is not required to be unitary or to have finite order, while in this paper, the multiplier systems of all modular forms that occur are unitary and have finite orders. In addition, it seems that the modular forms of rational weights that occur in this paper can only be regarded as modular forms on non-congruence subgroups of  with trivial character. As a comparison, Freitag and Hill <cit.> have recently constructed modular forms of weight 1/3 on SU(2,1) whose levels are certain congruence subgroups of SU(2,1). Furthermore, see <cit.> for harmonic weak Maass forms of real weight and their relations with weakly holomorphic modular forms.

First some notations. Let  be the group of all 2× 2 real matrices with positive determinants and  be the subgroup of  whose elements have determinant 1. Let D be a positive integer. Then we define the D-cover of  by

    D={abcdεDabcd∈, ε^D=1},

where a'b'c'd' is the matrix in  proportional to abcd and (c^'τ+d^')^1/D means a function of τ∈={z∈(z)>0}. The notation abcdεD is sometimes abbreviated to abcdε when D is understood. The composition is given by

    a_1b_1c_1d_1ε_1D·a_2b_2c_2d_2ε_2D
    
    = (([ a_1 b_1; c_1 d_1 ])([ a_2 b_2; c_2 d_2 ]),ε_1ε_2(c_1^'a_2^'τ+b_2^'/c_2^'τ+d_2^'+d_1^')^1/D(c_2^'τ+d_2^')^1/D).

It can be verified straightforwardly that D with this composition is a group. One should note that different choices of holomorphic branches of (c^'τ+d^')^1/D lead to different D-covers. We choose the following branch throughout:

    z^r=exp(rlog z),   - < (log z) ≤.

It is possible to make D a topological covering of  and hence D becomes a Lie group whose smooth structure is transposed from  using the above covering map (cf. <cit.>). The motivation for using D-covers is that when dealing with modular forms of weight k∈1/D, the multiplier systems become group characters on D-covers.

Let G be a subgroup of ; then by GD we understand the preimage of G under the natural projection D→, abcdε↦abcd. For γ∈, let γ denote (γ,1)∈D. (The cover index D should be inferred from the context.) In addition, we set G={g·{±1001} g∈ G}, which is a subgroup of =/{±1001}.

The group  is the set of 2× 2 integral matrices of determinant 1 and is known as the full modular group. We also need the congruence subgroup

    Γ_0(N)={abcd∈ c ≡ 0  N},

where N is a positive integer called the level. It is well known that the index [Γ_0(N)]=N∏_p | N(1+1/p) where p denotes a prime (cf. <cit.>). The matrices 1001, 1101, 0-110 and N001 are denoted by I, T, S, B_N respectively.

Let ℳ() denote the field of meromorphic functions on the upper half plane  and k∈1/D. Define the slash operator of weight k by

    f|_kabcdε(τ)=(ε√(c^'τ+d^'))^-Dkf(aτ+b/cτ+d),

where f∈ℳ(), abcdε∈D and τ∈. It is immediate that D acts on ℳ() on the right via the slash operator of weight k. Let G be a subgroup of  and χ be a character[A character always means a unitary linear character, that is, a group homomorphism to the group of complex numbers of absolute value 1.] on GD. We say f transforms like a modular form of weight k and with multiplier system (or with character) χ if f|_kγ=χ(γ)f for any γ∈GD. If this is the case, and [ G]<+∞ and χ(γ) has finite order for any γ∈GD, then for abcd∈ we have

    f|_kabcd(τ)=∑_n∈ m^-1c_nq^n,

where q=τ=exp(2τ) and m∈Z1 provided that there is Y_0>0 (depending on abcd) such that f|_kabcd has no poles on _Y_0={τ∈(τ)>Y_0}. Moreover, the series converges normally on _Y_0. To see this, note that there exists m∈Z1 such that χ(abcdT^mabcd^-1)=1 and hence f|_kabcd(τ+m)=f|_kabcd(τ). Therefore, the desired expansion follows from Fourier's theorem or Laurent's theorem. Define the order _∞(f|_k(abcd,ε)) to be the least n such that c_n ≠ 0. If the expansion does not exist (that is, f|_k(abcd,ε) has a nonisolated singularity at infinity) then the order is undefined; if the expansion holds but for any n_0∈ there exists n<n_0 such that c_n ≠ 0 then define _∞(f|_k(abcd,ε))=-∞; if f|_k(abcd,ε) is identically zero then define _∞(f|_k(abcd,ε))=+∞.



Let D∈Z1 and k∈1/D. Let G be a subgroup of  of finite index and χGD→^× be a character of finite order[If χ is not of finite order as considered by Knopp and Mason <cit.>, this definition also makes sense.] (that is, χ(γ) has finite order for any γ∈GD). Let f∈ℳ(). Then we say f is a meromorphic modular form of weight k for G with character χ if f|_kγ=χ(γ)f for any γ∈GD and _∞(f|_kγ)>-∞ for any γ∈D. Suppose this is the case. Then we say f is a weakly holomorphic modular form if it has no poles on ; it is a modular function if the weight k=0; it is a modular form (cusp form respectively) if it is weakly holomorphic and _∞(f|_kγ)≥ 0 (>0 respectively) for any γ∈D. The vector spaces (over ) of modular forms and of cusp forms are denoted by M_k(G,χ) and S_k(G,χ) respectively. When χ is trivial (which means k∈2 and χ always takes the value 1), we let M_k(G)=M_k(G,χ) and S_k(G)=S_k(G,χ).


Note that the cover index D can be recovered from the domain of χ and M_k(G,χ) is also denoted by M_k(GD,χ) when needed.



Suppose there exists a nonzero meromorphic modular form f of weight k for G with character χ. Then we have the following facts:

  
  * χ(I,1/D)=-k.
  
  * If -I∈ G, then χ(-I)=-k/2.
  
  * If D' is another positive integer with k∈1/D' and D'| D, then χGD→^× descends to a character on GD', i.e., χ can be factored as χ=χ_1p_D,D' where p_D,D'GD→GD' is the natural projection (γ,ε)↦(γ,ε^D/D') and χ_1 is a character on GD' (which is unique).

The proof is straightforward by considering slash operators acting on f of which we omit the details. Motivated by the first two facts, we call a character χ on GD with the properties χ(I,1/D)=-k and χ(-I)=-k/2 if -I∈ G a multiplier system for G of weight k of cover index D. Note that if -I∈ G, then the second property implies the first one. Finally, note that the third fact still holds when we require that χ is a multiplier system for G of weight k of cover index D even if we can not find a nonzero form f.




If f is a meromorphic modular form of weight k for G with character χ, and g is a meromorphic modular form of weight k' for G with character χ' with k,k'∈1/D, then it is immediate that f· g is a meromorphic modular form of weight k+k' for G with character χ·χ'. Moreover, if f and g are both modular forms (cusp forms respectively), then so is f· g. 


One of the aims of this paper is to give explicit formulas concerning _ M_k(Γ_0(N),χ) and _ S_k(Γ_0(N),χ) where k∈ and χ is a character on Γ_0(N)D of some kind which we explain now.

We need the Dedekind eta function and its logarithm, which are defined by

    η(τ)   =q^1/24∏_n∈Z1(1-q^n),  τ∈, 
    logη(τ)   =logη()+∫_^τη'(z)/η(z) z,

where logη() is the real logarithm. The transformation equations of logη under  are obtained by Dedekind (cf. <cit.>):

    logη(aτ+b/cτ+d)-logη(τ)=2(a+d/24c+1/2s(-d,c)-1/8)+1/2log(cτ+d),

where abcd∈ with c>0 and s(-d,c) is the Dedekind sum

    s(h,k)=∑_r  k((r/k))·((hr/k)),   h∈, k∈Z1.

In the above definition, ((x))=x-[x]-1/2 if x∈∖ and ((x))=0 if x∈.
Set

    Ψ   →
    abcd   ↦a+d/c+12s(-d,c)-3,       c>0; 
    a+d/c+12s(d,-c)+3,       c<0; 
    
    b,                               c=0, a>0; 
    
    -b-6,                            c=0, a<0.

Fix a positive integer D. Then the transformation equations of η follow from that of logη and can be expressed as η|_1/2(γ,ε)=χ_η(γ,ε)η, where χ_η is the multiplier system for  of weight 1/2 of cover index 2D defined by

    χ_η(γ,ε)=ε^-DΨ(γ)/24,  (γ,ε)∈2D.

The fact that Ψ()⊆ (when we give (<ref>) this is tacitly assumed) follows from the above formula with D=1, the fact  is generated by T and S and the fact χ_η(T)^24=χ_η(S)^24=1.

Let N,D∈Z1. By an eta-quotient of level N and cover index D we understand a product f(τ)=∏_0<n| Nη(nτ)^r_n with r_n∈ such that D· r_n∈2 for any n| N. Thus an eta-quotient of cover index 2 is just an ordinary eta-quotient whose exponents are integers. The fractional powers are defined by η(nτ)^r_n=exp(r_nlogη(nτ)) (cf. (<ref>)). Since η has no poles on , the eta-quotient f is a weakly holomorphic modular form of weight k=1/2∑_n| Nr_n for Γ_0(N)D with character

    χΓ_0(N)D   →^×
    abcdε   ↦ε^-Dk1/24∑_n | Nr_nΨabnc/nd.

The above formula can be proved using (<ref>), Remark <ref> and the fact η(nτ)=n^-1/4η|_1/2B_n(τ). Note that η has no zeros on  according to its infinite product expansion. Therefore any eta-quotient has no zeros or poles on . For the order at infinity, we have (cf. <cit.>)

    _∞(f|_kabcd)=1/24∑_n | N(n,c)^2/nr_n,  abcd∈.

We recommend the reader to see <cit.> or <cit.> for more details on ordinary eta-quotients. If (<ref>) are always nonnegative for all abcd∈, we say f is a holomorphic eta-quotient.

We will investigate the dimensions _ M_k(Γ_0(N),χ) and _ S_k(Γ_0(N),χ) where k∈1/D and χ is the character of an eta-quotient of weight k, level N and cover index D in the next three sections.


The reader should be warned that when dealing with half-integral weights, our notation may differ from other authors', e.g. from the notation encountered in <cit.> and <cit.>. Let ψ be a Dirichlet character modulo 4N and let 2k∈Z1. The space M_0(4N, k, ψ) in Serre and Stark's notation is the same thing as M_k(Γ_0(4N), χ_1ψ_1) in our notation where χ_1 is the character of η(τ)^-4kη(2τ)^10kη(4τ)^-4k and ψ_1 is the character that maps abcdε to ψ(d). The character χ_1 is hidden in many authors' notation of spaces and it sometimes appears in their definitions of slash operators or modular transformations or automorphic factors.





§ DIVISORS OF MODULAR FORMS


Our method relies on the classical Riemann-Roch theorem, so we recall some elements here. By a compact Riemann surface X, we understand a compact connected Hausdorff topological space equipped with a maximal atlas {(U_i,ϕ_i)} (i.e., each U_i is open in X and ϕ_i is a homeomorphism from U_i onto an open set in ) such that when (U_i,ϕ_i) and (U_j,ϕ_j) overlap, the transition map ϕ_j∘ϕ_i^-1ϕ_i(U_i∩ U_j)→ϕ_j(U_i∩ U_j) is holomorphic. It can be shown that X is second-countable (cf. <cit.>). Therefore, according to the well known classification of compact connected topological surfaces (cf. eg. <cit.>) and the fact that Riemann surfaces are oriented, X is homeomorphic to the 2-sphere 𝕊^2 or to the connected sum of g copies of the real projective plane ℙ^2 with g∈Z1. The number g is called the genus of X. (If X is a sphere, then its genus is defined to be 0.)

A meromorphic differential ω on X is by definition a holomorphic differential on X∖ S where S is a finite subset of X such that if (U,ϕ U→ϕ(U)⊆) is any chart of X, then ω is of the form f(z) z on this chart where f(z) is a meromorphic function on ϕ(U) whose poles are exactly ϕ(U∩ S). The basic feature of a meromorphic differential is the transformation equation between different charts: if (V,ψ) is another chart that overlaps (U,ϕ) and ω=g(w) w on (V,ψ), then we have

    f(z)=g(ψ∘ϕ^-1(z))ψ∘ϕ^-1(z)/ z,   z∈ϕ(U∩ V).

The set of all meromorphic differentials on X is denoted by 𝒦(X) which is obviously a complex vector space. Let ℳ(X) denote the complex vector space of all meromorphic functions on X as in Section <ref>. If there exists a nonzero ω_0∈𝒦(X), then the map ℳ(X)→𝒦(X) that sends f to f·ω_0 is a -linear isomorphism. Such ω_0 actually exists according to a fundamental result in the theory of Riemann surfaces (cf. <cit.>). As a consequence, nonzero meromorphic functions always exist on any compact Riemann surface.

By a (-valued) divisor on X, we understand a function X→ with finite support. We often write a divisor D as a formal sum ∑_x∈ Xd_x·(x) where d_x=D(x). The term “divisor” traditionally refers to a -valued divisor but in this paper to a -valued divisor. The sets of all -valued divisors and of all -valued divisors are denoted by 𝒟(X) and 𝒟_(X) respectively. Equipped with the pointwise addition, they are abelian groups. We define the degree of D∈𝒟_(X) by (D)=∑_x∈ XD(x), which is well-defined since the sum is actually a finite sum. Moreover, define the floor function as follows: for D∈𝒟_(X), set [D]=∑_x∈ X[d_x]·(x), where [d_x] is the greatest integer that does not exceed d_x. Finally, for two divisors D_1 and D_2, we say D_1≥ D_2 if D_1(x) ≥ D_2(x) far any x∈ X, and we say D_1 > D_2 if D_1 ≥ D_2 but D_1≠ D_2.

Generally speaking, with any meromorphic section of a holomorphic line bundle over X one can associate a -valued divisor. Specifically, if f∈ℳ(X), then define ÷(f)=∑_x∈ Xd_x·(x) with d_x being the least exponent in the Laurent expansion of f at x in any chart. For ω∈𝒦(X) we define ÷(ω) in a similar manner.

The compact Riemann surfaces we need are the modular curves X_G where G is a finite index subgroup of  which we describe now. Let ^*=∪ where  is the projective line over  which can be identified with  together with a point ∞=1/0. The group  acts on  on the left via abcd·(p/q)=ap+bq/cp+dq. We endow ^* with the topology generated by the usual topology of  and the sets

    {γττ∈_Y_0∪{∞}},  γ∈, Y_0>0,

where _Y_0={τ∈(τ)>Y_0}. It is immediate that ^* with this topology is a Hausdorff space[The space ^* is not locally compact and hence can not be a topological manifold.],  is an open subset of ^* and G acts on ^* (on the left) by homeomorphisms. Let G\^* be the orbit space (endowed with the quotient topology) and p^*→ G\^* be the quotient map. It can be shown that G\^* is Hausdorff (cf. eg. <cit.>) and when [ G]<+∞ G\^* is compact (cf. eg. <cit.>). Hereafter we always assume that [ G]<+∞. Write G\^*=G\∪ G\, a disjoint union. Then G\ is open in G\^* and G\ is a finite set since  acts transitively on  and hence G\≤[ G]. 
The orbits in G\ are called cusps of G\^* and by abuse of language, the points in  are also called cusps. The width of a cusp G· x∈ G\ is defined by w_Gx=w_x=[_xG_x] which is independent of the choice of the representative x. We have (cf. <cit.>)

    [G]=∑_Gx∈ G\w_Gx.

We also need the notion of elliptic points. It is well known that the stabilizer of each point in {γγ∈} under the action of /{± I} is a cyclic group of order 2, the stabilizer of each point in {γργ∈} is a cyclic group of order 3 where ρ=-1+√(3)/2 and the stabilizer of any other point in  is trivial. Thus for τ∈, G_τ=1, 2 or 3 since G_τ divides _τ. If G_τ=2 (3 respectively) then we call p(τ) an elliptic point for G\^* of period 2 (3 respectively). By abuse of language, we also call the number τ an elliptic point if p(τ) is.

The description and verification of the atlas on G\^* that turns it into a compact Riemann surface are rather technical and tedious but it is basic and well known (cf. eg. <cit.>). Let X_G denote the resulting compact Riemann surface. We will mainly use X_Γ_0(N) hereafter so set X_0(N)=X_Γ_0(N) for simplicity. Let g be the genus of X_G. Then by applying the Riemann-Hurwitz ramification formula to the holomorphic map X_G→ X_, G· z ↦· z, noticing that the degree of this map equals m=[G] and the genus of X_ is 0, we obtain (cf. e.g. <cit.> for the case G is a congruence subgroup)

    g=1+m/12-ε_2/4-ε_3/3-ε_∞/2,

where ε_2 and ε_3 are the numbers of elliptic points of X_G of period 2 and 3 respectively and ε_∞ is the number of cusps of X_G. More formally,

    ε_2   =#{G· z∈ G\G_z=2},
    ε_3   =#{G· z∈ G\ρG_z=3},  ρ=-1+√(3)/2,
    ε_∞   =#G\.

For G=Γ_0(N), it is known that

    m   =[Γ_0(N)]=[Γ_0(N)]=N∏_p| N(1+1/p),
    ε_2   =∏_p | N(1+-4p)     if  4∤ N,
    
    0     if  4| N,
    
    ε_3   =∏_p | N(1+-3p)     if  9∤ N,
    
    0     if  9| N,
    
    ε_∞   =∑_d| Nϕ(d,N/d),

where p denotes a prime, ϕ(d,N/d)=ϕ((d,N/d)), and ϕ is the Euler function. For proofs, see <cit.> and <cit.>.

Now let us associate a -valued divisor to any meromorphic modular form of rational weight. For integral or half-integral weights, this association is the same as the ordinary one[But it is different from the one described in <cit.>] (cf. <cit.>). Our treatment has the feature that one need not distinguish between regular and irregular cusps.


Let G be a subgroup of  of finite index, D∈Z1 and k∈1/D. Let χ be a multiplier system for G of weight k of cover index D (not assumed to be of finite order). Let f be a nonzero meromorphic modular form of weight k for G with character χ. We define its divisor ÷(f)=∑_x∈ X_G÷_x(f)·(x)∈𝒟_(X_G) as follows:

  
  * If τ∈, then set ÷_Gτ(f)=n_0/G_τ where n_0 is the integer such that f(z)=∑_n≥ n_0c_n(z-τ)^n, c_n_0≠ 0.
  
  * If x∈ and γ∈ satisfying γ(∞)=x, then set ÷_Gx(f)=_∞(f|_kγ)· w_Gx.


One can verify that ÷(f) is well-defined, which is not totally trivial but straightforward. The usefulness of this notion relies on the following two simple facts:

  
  * If g is another nonzero meromorphic modular form of weight k'∈1/D for G with character χ', then we have ÷(fg)=÷(f)+÷(g) in 𝒟_(X_G) (cf. Remark <ref>).
  
  * If k=0 and χ is the trivial character, then f descends to a meromorphic function f on X_G. We have ÷(f)=÷(f) where ÷(f) is the usual divisor of a meromorphic function.

Note that the factors 1/G_τ and w_Gx in Definition <ref> are chosen to let the latter fact hold (to see this one must dive into the atlas of X_G which we have omitted). Also note that the former fact holds because the factors 1/G_τ and w_Gx remain unchanged for all k.

Similarly, we have ÷(f/g)=÷(f)-÷(g) and in particular ÷(1/g)=-÷(g).

A fundamental theorem on modular forms of rational weight is the following one, which in cases of integral or half-integral weights is sometimes called the valence formula.


Let us use the notation of Definition <ref> and set m=[G]. Then we have

    (÷ f)=1/12mk.



Suppose that =⋃_1≤ j≤ mGγ_j is a disjoint union where γ_j∈ and γ_j=γ_j·{± I}. Set g=∏_1 ≤ j ≤ mf|_kγ_j. Then g|_mkγ=c_γ g for any γ∈D where c_γ∈^×. Since g is not identically zero, the map γ↦ c_γ is a linear character on D. According to a presentation of D (cf. <cit.>) the order of this character divides 12D and hence g^12D transforms like a modular form of weight 12Dmk for  with trivial character. Since f is meromorphic at cusps, so is g^12D. Therefore g^12D is a meromorphic modular form of weight 12Dmk for  with trivial character. Applying the usual valence formula for even weights (cf. <cit.>) we obtain (÷ g^12D)=Dmk. It remains to prove that 12D(÷ f)=(÷ g^12D), which is equivalent to (÷ f)=(÷ g). We now prove a stronger assertion, i.e.

    ÷_x(g)=∑_y∈ p^-1(x)÷_y(f)

for any x∈ X_, where p X_G→ X_ is the natural projection G·τ↦·τ. In the case that x∈\, we write x=·τ and set w_τ=[_τG_τ] (similar to the notion of the width of a cusp in the group-theoretical aspect). Notice that _τ f|_kγ_j=_γ_jτf where _τ f is the integer n_0 such that lim_z→τf(z)(z-τ)^-n_0 is nonzero. Thus we have

    ÷_x(g)=_τ^-1·∑_1 ≤ j ≤ m_γ_jτ f.

The underlying set of the multiset {G·γ_jτ 1 ≤ j ≤ m} equals p^-1(x) and the multiplicity of each element G·γ_jτ equals w_γ_jτ. It follows that

    ÷_x(g)=_τ^-1·∑_y=G·τ∈ p^-1(x)w_τ·_τ f=∑_y∈ p^-1(x)÷_y(f),

i.e., the desired assertion (<ref>) holds. In the other case x∈\, we have x=·∞ and its width is 1. Therefore

    ÷_x(g)=∑_1 ≤ j ≤ m_∞(f|_kγ_j).

The underlying set of the multiset {G·γ_j∞ 1 ≤ j ≤ m} equals p^-1(x) and the multiplicity of each element G·γ_j∞ equals w_γ_j∞. It follows that

    ÷_x(g)=∑_y=G·γ∞∈ p^-1(x)w_γ∞_∞(f|_kγ)=∑_y∈ p^-1(x)÷_y(f),

i.e., the desired assertion (<ref>) holds. This concludes the proof.



The above theorem holds even when χ is non-unitary and the proof remains unchanged. For instance, one can apply it to the generalized modular forms constructed in <cit.>.


In the rest we need to know information about the divisor of an eta-quotient times a product of Eisenstein series on . We let

    E_4(τ)   =1+240∑_n∈Z1σ_3(n)q^n,
    
    E_6(τ)   =1-504∑_n∈Z1σ_5(n)q^n,

where σ_k(n)=∑_d| nd^k. It is well known that E_4∈ M_4() and E_6∈ M_6().


Let N∈Z1, r_n∈ for any n| N and t_1,t_2∈. Let χ be the character (<ref>) where D∈Z1 is chosen so that Dr_n∈2 for any n | N. Set

    f(τ)=∏_n | Nη(nτ)^r_n· E_4(τ)^t_1· E_6(τ)^t_2.

Then f is a nonzero meromorphic modular form of weight k=1/2∑_nr_n+4t_1+6t_2 for Γ_0(N) with character χ and

    ([÷ f])=∑_c| Nϕ(c,N/c)·[N/24(N,c^2)∑_n | N(n,c)^2/nr_n]+(t_1/3+t_2/2)m-{t_2/2}ε_2-{t_1/3}ε_3,

where m,ε_2 and ε_3 are given in (<ref>), (<ref>) and (<ref>).


The fact that f is a nonzero meromorphic modular form follows from Remark <ref>. It remains to compute ([÷ f]). The divisors of E_4 and E_6 on X_Γ_0(N) are known (cf. eg. <cit.>) as follows:

    ÷_x(E_u)=
    1/e_τ    if  x=Γ_0(N)·τ∈Γ_0(N)\·τ_0,
    
    0     otherwise,

where u=4 or 6, e_τ=Γ_0(N)_τ and τ_0=ρ=-1+√(3)/2 if u=4; τ_0= if u=6. On the other hand, if we set f_0(τ)=∏_n | Nη(nτ)^r_n, then ÷_x(f_0)=0 for x∈Γ_0(N)\ since f_0 has no poles or zeros on . For x∈Γ_0(N)\, we can find abcd∈ such that x=Γ_0(N)·abcd∞. It is known that w_x=N/(N,c^2) (cf. <cit.>). It follows from this and (<ref>) that

    ÷_x(f_0)=N/24(N,c^2)∑_n | N(n,c)^2/nr_n.

Therefore

    ([÷ f])=∑_x∈Γ_0(N)\ρ[t_1·÷_x(E_4)]+∑_x∈Γ_0(N)\[t_2·÷_x(E_6)]+∑_x∈Γ_0(N)\[÷_x(f_0)],

where the three sums in the right-hand side are denoted by S_1, S_2 and S_3 respectively hereafter.
To compute S_1, let =∪_1 ≤ j ≤ mΓ_0(N)γ_j be a disjoint union[Since -I∈Γ_0(N), this is equivalent to =∪_1 ≤ j ≤ mΓ_0(N)γ_j, and as well to =∪_1 ≤ j ≤ mΓ_0(N)γ_j both of which are disjoint unions.]. Then the underlying set of the multiset {Γ_0(N)·γ_jρ 1 ≤ j ≤ m} equals Γ_0(N)\ρ and the multiplicity of each element Γ_0(N)·γ_jρ equals _γ_jρ/Γ_0(N)_γ_jρ=3/e_γ_jρ. Hence we have

    S_1=1/3∑_1 ≤ j ≤ m e_γ_jρ=1[t_1/e_γ_jρ] + ∑_1 ≤ j ≤ m e_γ_jρ=3[t_1/e_γ_jρ]=mt_1/3-{t_1/3}ε_3.

Similarly, S_2=mt_2/2-{t_2/2}ε_2. For S_3, we need a complete set of representatives of Γ_0(N)\, i.e., the set of a/c∈ where c| N and for each a_0 such that 0 ≤ a_0 < (c,N/c) with (a_0,c,N/c)=1, a is chosen to satisfy a≡ a_0  (c,N/c) and (a,c)=1 (cf. <cit.>). It follows from this and (<ref>) that

    S_3=∑_c| Nϕ(c,N/c)·[N/24(N,c^2)∑_n | N(n,c)^2/nr_n].

Inserting the expressions for S_1, S_2 and S_3 into (<ref>) gives the desired formula.





§ THE MAIN THEOREM: DIMENSION FORMULAS FOR RATIONAL WEIGHTS


We recall the Riemann-Roch theorem here for proving the main theorem. Let X be a compact Riemann surface of genus g and D∈𝒟_(X). Then the Riemann-Roch space is defined by ℒ(D)={f∈ℳ(X)÷(f)≥-D} with the convention that 0∈ℒ(D). It is immediate that ℒ(D) is a complex vector space and that ℒ(D)=ℒ([D]). Similarly we define 𝒦(D)={ω∈𝒦(X)÷(ω)≥ D}∪{0}. Then the Riemann-Roch theorem states that

    _ℒ(D)=_𝒦(D)+(D)-g+1,

provided that D is integral, i.e., D∈𝒟(X). Note that it is tacitly understood that _ℒ(D)<+∞. For a proof based on the existence theorems for harmonic functions on Riemann surfaces, see <cit.>. Set D=0; we find that _𝒦(0)=g, that is, the dimension of the space of all holomorphic differentials is g. Let ω_0 be any nonzero meromorphic differential (necessarily exists) and K=÷(ω_0). Then ℒ(K-D)→𝒦(D), f↦ f·ω_0 is a -linear isomorphism. Now set D=K in (<ref>); we find that (K)=2g-2. Finally, if D∈𝒟(X) and (D)>2g-2, then (K-D)<0 and hence ℒ(K-D)=𝒦(D)={0} since the degree of a nonzero meromorphic function is 0. It follows that

    _ℒ(D)=(D)-g+1,  if  D∈𝒟(X), (D)>2g-2.


Our proof of the main theorem rests on the following general lemma, which is the key point for using the Riemann-Roch theorem.


Let G be a subgroup of  of finite index, D∈Z1, k∈1/D and vG^D→^× be a character. Let p→ G\ be the natural projection. Suppose there exists a nonzero meromorphic modular form f_0 of weight k for group G with character v. Then the map

    ℒ([÷ f_0])    → M_k(G,v)
    
    f    ↦ (f|_G\)∘ p · f_0

is a -linear isomorphism. Moreover, let

    R={x∈ G\÷_x(f_0)∈}.

Then the map

    ℒ([÷ f_0]-∑_x∈ R(x))    → S_k(G,v)
    
    f    ↦ (f|_G\)∘ p · f_0

is also a -linear isomorphism.


Let f∈ℒ([÷ f_0]) be arbitrary. Since p is holomorphic, (f|_G\)∘ p is a modular function for group G with trivial character. Thus (f|_G\)∘ p · f_0 is a meromorphic modular form of weight k for group G with character v by Remark <ref>. Now assume f≠ 0 and let x∈ X_G be arbitrary. Since f∈ℒ([÷ f_0]) we have ÷_x((f|_G\)∘ p)=÷_x(f)≥ -÷_x(f_0). It follows that

    ÷_x((f|_G\)∘ p · f_0)=÷_x((f|_G\)∘ p)+÷_x(f_0) ≥ 0.

Hence (f|_G\)∘ p · f_0∈ M_k(G,v). It is obvious that the considered map is a -linear injection. To prove the surjectivity, let 0≠ g∈ M_k(G,v) be arbitrary, then by Remark <ref> g/f_0 is a modular function for group G with trivial character and hence it descends to some nonzero f∈ℳ(X_G). For any x∈ X_G we have

    ÷_x(f)=÷_x(g/f_0)=÷_x(g)-÷_x(f_0)≥-÷_x(f_0).

Therefore f∈ℒ(÷ f_0)=ℒ([÷ f_0]) and the image of f is g from which the surjectivity follows.

The assertion on the second map can be proved in a similar manner and the only thing that need to explain is that (f|_G\)∘ p · f_0∈ S_k(G,v) if and only if f∈ℒ([÷ f_0]-∑_x∈ R(x)). If 0≠ f∈ℒ([÷ f_0]-∑_x∈ R(x)) and x∈ R, then

    ÷_x((f|_G\)∘ p · f_0)=÷_x(f)+÷_x(f_0) ≥ 1-[÷_x(f_0)]+÷_x(f_0)>0.

On the other hand, if x∉R then -[÷_x(f_0)]+÷_x(f_0)>0 and hence

    ÷_x((f|_G\)∘ p · f_0)=÷_x(f)+÷_x(f_0) ≥ -[÷_x(f_0)]+÷_x(f_0)>0.

It follows that (f|_G\)∘ p · f_0∈ S_k(G,v). The converse can be proved similarly.


Now we state and prove the main theorem.


Let N∈Z1, r_n∈ for any n | N and t∈Z0. Let D be a positive integer such that Dr_n∈2 for any n | N. Let χΓ_0(N)D→^× be the character (<ref>) and m, ε_2, ε_3 are given by (<ref>), (<ref>), (<ref>) respectively. Set k=1/2∑_n r_n+2t and

    x_c=∑_n| NN/(N,c^2)·(n,c)^2/nr_n

with c| N. Suppose either

    ∑_c| Nϕ(c,N/c)·([x_c/24]+1)>0,  t ≥ 1

or

    k>2-6/mε_2-8/mε_3-12/m∑_c| Nϕ(c,N/c)·(1-{x_c/24}), t=0.

Then we have

    _ M_k(Γ_0(N), χ)=k-1/12m+(1/4-{t/2})ε_2+(1/3-{-t/3})ε_3
    
    +∑_c| Nϕ(c,N/c)·(1/2-{x_c/24}).



Set

    f_0(τ)=∏_n | Nη(nτ)^r_n· E_4(τ)^-t· E_6(τ)^t.

According to Proposition <ref> f_0 is a nonzero meromorphic modular form of weight k for group Γ_0(N) with character χ. Hence _ M_k(Γ_0(N), χ)=_ℒ([÷ f_0]) by Lemma <ref>. Using Proposition <ref> again we find that

    ([÷ f_0])=∑_c| Nϕ(c,N/c)·[x_c/24]+t/6m-{t/2}ε_2-{-t/3}ε_3.

We now prove either (<ref>) or (<ref>) implies that ([÷ f_0])>2g-2 where g is the genus of X_Γ_0(N) (cf. (<ref>)). First suppose (<ref>) holds. If N=1, then g=0, r_1≥ 0 and hence

    ([÷ f_0])=[r_1/24]+t/6-{t/2}-{-t/3}>-2=2g-2.

If N ≥ 2, set

    γ(t)=t/6m-{t/2}ε_2-{-t/3}ε_3.

We have

    γ(t+1)-γ(t)≥1/6m-1/2ε_2-2/3ε_3=2g-2+ε_∞

for t ≥ 1. Since N ≥ 2 we have ε_∞≥ 2 and hence γ(t+1)-γ(t)≥ 0. Therefore ([÷ f_0]) as a function of t∈Z1 is increasing which means we need only to prove

    ∑_c| Nϕ(c,N/c)·[x_c/24]+1/6m-1/2ε_2-2/3ε_3>2g-2.

This is actually equivalent to (<ref>) according to (<ref>) and (<ref>). Thereby we have shown that (<ref>) implies ([÷ f_0])>2g-2. Next suppose (<ref>) holds, which is equivalent to

    1/12mk>1/6m-1/2ε_2-2/3ε_3-ε_∞+∑_c| Nϕ(c,N/c){x_c/24}=2g-2+∑_c| Nϕ(c,N/c){x_c/24}.

Applying Theorem <ref> and (<ref>) to f_0 (noting t=0 here) and using the complete set of representatives of Γ_0(N)\ described in the proof of Proposition <ref> we find that

    1/12m·1/2∑_nr_n=(÷ f_0)=∑_c | Nϕ(c,N/c)·x_c/24.

Inserting this into (<ref>) we obtain ([÷ f_0])>2g-2 as required.

Thus, in both cases we can apply (<ref>) to D=[÷ f_0], X=X_Γ_0(N). It follows that

    _ M_k(Γ_0(N), χ)=_ℒ([÷ f_0])=([÷ f_0])-g+1.

Inserting (<ref>), (<ref>) and (<ref>) into the above identity gives the desired formula.



If neither (<ref>) nor (<ref>) holds, then the difference between the left-hand side and the right-hand side of (<ref>) equals _𝒦([÷ f_0]) according to (<ref>). It follows that in all cases we have

    _ M_k(Γ_0(N), χ)≥k-1/12m+(1/4-{t/2})ε_2+(1/3-{-t/3})ε_3
    
    +∑_c| Nϕ(c,N/c)·(1/2-{x_c/24}),

where the right-hand side is always an integer. We can derive an upper bound[This upper bound is still valid if Γ_0(N) is replaced by any subgroup G of finite index in .] for _ M_k(Γ_0(N), χ) as well. Suppose that k≥ 0. (When k<0 M_k(Γ_0(N), χ)={0} by Theorem <ref>.) Then

    _ M_k(Γ_0(N), χ) ≤[1/12mk]+1,

for if f_1,…,f_j∈ M_k(Γ_0(N), χ) with j>[1/12mk]+1, then by linear algebra we can find a nontrivial linear combination g=∑_n c_nf_n such that ÷_x(g)≥[1/12mk]+1 where x∈Γ_0(N)\ is not an elliptic point. If g ≠ 0 then applying Theorem <ref> to g we reach a contradiction. Hence g=0 which means f_1,…,f_j are linearly dependent.


We can derive a formula for _ S_k(Γ_0(N), χ) as well, using the second map in Lemma <ref>. However, it may happen that for certain r_n and t (<ref>) holds while the corresponding formula for _ S_k(Γ_0(N), χ) is not applicable. This happens precisely when ([÷ f_0])>2g-2 but ([÷ f_0])-R≤2g-2. Besides, when k>2, we always have _ S_k(Γ_0(N), χ)=_ M_k(Γ_0(N), χ)-R which is a well known fact when k is integral and χ is induced by a Dirichlet character. In the rational weight and arbitrary multiplier system case, this holds as well since the Petersson inner product can also be defined. The details are omitted here.





§ ORDER-CHARACTER RELATIONS


Let us assume (<ref>) holds and consider Theorem <ref>. The sequence (x_c)_c| N occurs in the formula for _ M_k(Γ_0(N),χ) and is determined by the sequence (r_n)_n | N via (<ref>). Note that (r_n)_n | N represents an eta-quotient (or its character) and (x_c)_c| N represents the orders of this eta-quotient at cusps (or its divisor). Therefore (<ref>) can be regarded as a map that sends the character represented by (r_n)_n | N to the orders at cusps (x_c)_c| N. In some potential applications, the sequence of orders (x_c)_c| N is first given, e.g., when one want to known _ M_k(Γ_0(N),χ) for fixed N and k, and then one works out (r_n)_n | N via (<ref>). Motivated by this, we need an inverse formula of (<ref>). This formula is an enhancement of a statement in <cit.> and is equivalent to <cit.>.

Let us fix some notations in linear algebra. For N∈Z1, 𝒟(N) denotes the set of positive divisors of N and σ_0(N)=𝒟(N). Let V be a -vector space of dimension σ_0(N), then a 𝒟(N)-indexed basis of V is a sequence (𝐛_n)_n∈𝒟(N) so that {𝐛_n} is a basis of V. Let W be a -vector space of dimension σ_0(M) with a 𝒟(M)-indexed basis (𝐛'_m)_m | M where M is another positive integer and let f V→ W be a linear map. The matrix of f with respect to the pair ((𝐛_n),(𝐛'_m)) is the 𝒟(M)×𝒟(N)-indexed sequence (a_m,n)_m| M,n| N  with the property f(𝐛_n)=∑_m | Ma_m,n𝐛'_m. All linear algebra machinery works for such kind of matrices. The difference is just the underlying index sets: we use 𝒟(N) and the ordinary one is {1,2,…,σ_0(N)}. We call the matrix (a_m,n)_m| M,n| N a 𝒟(M)×𝒟(N)-indexed matrix and when M=N a 𝒟(N)-indexed matrix.

Let N_1,N_2∈Z1 such that (N_1,N_2)=1 and let (a_c_1,n_1)_c_1,n_1| N_1, (b_c_2,n_2)_c_2,n_2| N_2 be 𝒟(N_1)-indexed and 𝒟(N_2)-indexed matrices respectively. Define

    (a_c_1,n_1)_c_1,n_1| N_1⊗(b_c_2,n_2)_c_2,n_2| N_2=(a_c_1,n_1· b_c_2,n_2)_c_1c_2,n_1n_2| N_1N_2.

It is immediate that this is a well defined -bilinear map from ^𝒟(N_1)×𝒟(N_1)×^𝒟(N_2)×𝒟(N_2) to ^𝒟(N_1N_2)×𝒟(N_1N_2). Moreover it is universal among all bilinear maps on ^𝒟(N_1)×𝒟(N_1)×^𝒟(N_2)×𝒟(N_2) and hence is a tensor product. We call it the Kronecker product.

Following Bhattacharya <cit.> we define A_N=(N/(N,c^2)·(n,c)^2/n)_c,n| N so that (<ref>) becomes x_c=∑_n| NA_N(c,n)r_n. It is immediate that A_N_1N_2=A_N_1⊗ A_N_2 whenever (N_1,N_2)=1.


Let N∈Z1, (x_c)_c| N and (r_n)_n| N be in ^𝒟(N). Then the following two relations are equivalent:

    x_c   =∑_n| NN/(N,c^2)·(n,c)^2/nr_n,  ∀ c| N, 
    
    r_n   =1/N∏_p | Np/p^2-1·∑_c| N(∏_p^α∥ Nb_p,α(v_p(n),v_p(c)))x_c,  ∀ n| N,

where p denotes a prime, v_p(n) is the integer β such that p^β∥ n and

    b_p,α(i,j)=
    p    if  i=j=0 or α,
    
    (p^2+1)p^min(i,α-i)-1   if  0<i=j<α,
    
    -p^min(j,α-j)   if i-j=1,
    
    0    otherwise.


In another words, we have

    A_N^-1(n,c)=1/N∏_p | Np/p^2-1·∏_p^α∥ Nb_p,α(v_p(n),v_p(c)).

This proposition is due to Bhattacharya (cf. <cit.>).

Suppose N_1,N_2∈Z1 such that (N_1,N_2)=1. Then A_N_1 and A_N_2 are both invertible if and only if A_N_1⊗ A_N_2 is and in this case (A_N_1⊗ A_N_2)^-1=A_N_1^-1⊗ A_N_2^-1 by basic properties of the Kronecker product. Therefore it is sufficient to prove that

    A_p^α^-1(p^i,p^j)=1/p^α-1(p^2-1)· b_p,α(i,j),

where p is a prime, α∈Z1 and 0≤ i,j≤α. This is equivalent to

    ∑_j=0^αA_p^α(p^i,p^j)· b_p,α(j,i)=(p^2-1)p^α-1, 
       ∑_j=0^αA_p^α(p^i_1,p^j)· b_p,α(j,i_2)=0   if  i_1≠ i_2.

Note that

    A_p^α(p^i,p^j)=p^α/(p^α,p^2i)·(p^j,p^i)^2/p^j=
    p^α-j     if i≤ j  and  i≤α/2,
    
    p^2i-j     if i≤ j  and  i>α/2,
    
    p^α-2i+j     if i≥ j  and  i≤α/2,
    
    p^j     if i≥ j  and  i>α/2.

Now we begin to prove (<ref>). In the case i=0, we have

    ∑_j=0^αA_p^α(p^i,p^j)· b_p,α(j,i)   =A_p^α(p^0,p^0)· b_p,α(0,0)+A_p^α(p^0,p^1)· b_p,α(1,0)
       =p^α· p + p^α-1·(-1)=(p^2-1)p^α-1

as required. In the case i=α, we have

    ∑_j=0^αA_p^α(p^i,p^j)· b_p,α(j,i)   =A_p^α(p^α,p^α-1)· b_p,α(α-1,α)+A_p^α(p^α,p^α)· b_p,α(α,α)
       =p^α-1·(-1) + p^α· p=(p^2-1)p^α-1

as required. In the case 0<i ≤α/2 we have

    ∑_j=0^αA_p^α(p^i,p^j)· b_p,α(j,i)
    
    =   A_p^α(p^i,p^i-1)· b_p,α(i-1,i)+A_p^α(p^i,p^i)· b_p,α(i,i)+A_p^α(p^i,p^i+1)· b_p,α(i+1,i)
    
    =   p^α-i-1·(-p^min(i,α-i)) + p^α-i· (p^2+1)p^min(i,α-i)-1+p^α-i-1·(-p^min(i,α-i))
    
    =   (p^2-1)p^α-1

as required. In the remaining case α/2<i<α we have

    ∑_j=0^αA_p^α(p^i,p^j)· b_p,α(j,i)
    
    =   A_p^α(p^i,p^i-1)· b_p,α(i-1,i)+A_p^α(p^i,p^i)· b_p,α(i,i)+A_p^α(p^i,p^i+1)· b_p,α(i+1,i)
    
    =   p^i-1·(-p^min(i,α-i)) + p^i· (p^2+1)p^min(i,α-i)-1+p^i-1·(-p^min(i,α-i))
    
    =   (p^2-1)p^α-1

which concludes the proof of (<ref>). We can prove (<ref>) in a similar manner which is tedious (there are so many cases) so we omit the details.



There is another useful formula concerning the matrix A_N:

    ∑_c| Nϕ(c,N/c)N/(N,c^2)(n,c)^2/n=N∏_p | N(1+1/p).

In another words, the Euclidean inner product of each column of A_N and (ϕ(c,N/c))_c| N is equal to N∏_p | N(1+1/p). This can be proved by applying Theorem <ref> to η(nτ). As a consequence, ∑_c| Nϕ(c,N/c)x_c=N∏_p | N(1+1/p)·∑_n| Nr_n.



We present some examples concerning dimensions of spaces of weight 2 for infinite many unitary characters. Suppose that N is square-free for simplicity. Let 𝐱=(x_c)_c| N∈^𝒟(N) such that ∑_c| Nx_c=0. Then (<ref>) with t=1 holds since ϕ(c,N/c)=1 in the current case. Thus (<ref>) holds. According to the above remark we have ∑_n| Nr_n=m^-1∑_c| Nx_c=0 and hence k=2. Therefore (<ref>) is equivalent to

    _ M_2(Γ_0(N), χ)=g-1+σ_0(N)+∑_c| N[x_c/24],

where g is the genus of X_Γ_0(N). The multiplier system χ is the character (<ref>) where (r_n)_n| N is determined by 𝐱 via (<ref>) which in the current case is

    r_n=∏_p| N(p^2-1)^-1·∑_c| N((-1)^#{p| N v_p(n)≠ v_p(c)}·∏_p| N, v_p(n)=v_p(c)p)x_c.

More generally, if N∈Z1 is not divisible by 16 and has no odd square factor except 1, then we have ϕ(c,N/c)=1 for any c| N as well and hence (<ref>) is still valid. However in this case, (<ref>) should be modified slightly at the prime factor 2.

Now we try to find out one-dimensional spaces among the above spaces. Note that

    g-1<_ M_2(Γ_0(N), χ)≤ g-1+σ_0(N),

so it is necessary g=0 or 1 and N>1. When g=0, since N∈Z2 is not divisible by 16 and has no odd square factor except 1, then N∈{2,3,4,5,6,7,8,10,12,13}. For such N _ M_2(Γ_0(N), χ)=1 if 𝐱 satisfies

    ∑_c| Nx_c=0,  ∑_c| N[x_c/24]=2-σ_0(N).

When g=1, we have N∈{11,14,15,17,19,20,21,24}. For such N, _ M_2(Γ_0(N), χ)=1 if 𝐱 satisfies

    ∑_c| Nx_c=0,  ∑_c| N[x_c/24]=1-σ_0(N).


Let us look at a very simple case N=11. Since σ_0(11)=2, _ M_2(Γ_0(11), χ)=1 if and only if x_1∉24 (of course x_11=-x_1). We have r_1=-r_11=x_1/10 so χ is the multiplier system of (η(τ)/η(11τ))^x_1/10 which descends to a unitary character on Γ_0(11). If x_1∈24, then _ M_2(Γ_0(11), χ)=2. There are two completely different subcases in this case: x_1∈120 and x_1∉120. If x_1∈120, then χ is trivial and the formula reads _ M_2(Γ_0(11))=2 which is well known (cf. <cit.>). Otherwise, if x_1∉120 then x_1∉10, which implies that χ is the character of an eta-quotient of rational exponents. To our best knowledge, the dimension formulas in such situation are new. Moreover, it seems that χ is a noncongruence subgroup of  so that our formulas may be applied to the theory of noncongruence modular forms.





§ THE CLASSIFICATION OF CHARACTERS INDUCED BY ETA-QUOTIENTS


In this section we give a complete classification of the characters of eta-quotients of level N and cover index 2 for any N∈Z1. The motivation is that, with this classification in hand, we can know exactly the set of χ for which _ M_k(Γ_0(N), χ) can be calculated using Theorem <ref>. Since we only deal with the cover index D=2, set Γ_0(N)=Γ_0(N)2 in this section.

Now we establish some notations. Each (r_n)_n| N∈^𝒟(N) represents an eta-quotient ∏_n| Nη(nτ)^r_n of level N and cover index 2 bijectively and (r_n)_n| N with ∑_nr_n=2k represents an eta-quotient of weight k. Thus, set

    S(N,k)={(r_n)_n| N∈^𝒟(N)∑_n| Nr_n=2k},   N∈Z1,  k∈1/2,

and S(N)=S(N,0).
Then ^𝒟(N)=∪_2k∈S(N,k) which is a disjoint union. In fact, S(N) is a submodule of ^𝒟(N) and each S(N,k) is a coset in ^𝒟(N)/S(N). The element (r_n)_n| N in ^𝒟(N) is customary to be written as 𝐫=∑_n| Nr_ne_n where {e_n n| N} is the standard basis of ^𝒟(N). Moreover, the character (<ref>) with D=2 is denoted by χ_𝐫 to emphasize its dependence upon 𝐫∈ S(N,k).

Two different vectors 𝐫, 𝐫' in S(N,k) may correspond to the same character χ_𝐫=χ_𝐫'. The following lemma of Newman <cit.> tells us precisely when this happens.


Let N∈Z1, k∈1/2. Let 𝐫=(r_n)_n| N and 𝐫'=(r'_n)_n| N be vectors in S(N,k). Then χ_𝐫=χ_𝐫' if and only if

    ∑_n| Nn·(r_n-r'_n)≡024,
    ∑_n| NN/n·(r_n-r'_n)≡024,
    ∏_2∤ r_n-r'_nn  is a perfect square.



This is a special case of the last assertion of <cit.>. Setting l=1 in that assertion and noting that χ_𝐫=χ_𝐫' if and only if they are I-compatible give the desired equivalence.


It seems that this method does not work in the case D≥3 since we can not find any generalized double coset operator (cf. <cit.>) between characters of eta-quotients of cover index D≥3.


As a corollary, for 𝐫=(r_n)_n| N∈ S(N), χ_𝐫 is trivial if and only if ∑_nn· r_n≡024, ∑_nN/n· r_n≡024 and ∏_2∤ r_nn is a perfect square. The set of all such 𝐫 is denoted by Z(N) which is a subgroup of S(N). We define

    𝒮(N,k)=S(N,k)/Z(N)={𝐫+Z(N)𝐫∈ S(N,k)},

and 𝒮(N)=𝒮(N,0).
Thus ^𝒟(N)/Z(N)=∪_2k∈𝒮(N,k), which is a disjoint union. Note that elements of 𝒮(N,k) are in one-to-one correspondence with characters of eta-quotients of level N, cover index 2 and weight k according to Lemma <ref>, so to classify such characters we need only to obtain a complete system of representatives of 𝒮(N,k). Moreover, the map 𝒮(N)→𝒮(N,k) that sends ∑_n| Nr_ne_n+Z(N) to 2ke_1+∑_n| Nr_ne_n+Z(N) is a bijection. Therefore, we have reduced our task to the study of 𝒮(N). (For the result, see Theorem <ref>.)


Let χ_𝐫Γ_0(N)→^× be a character given by (<ref>) where 𝐫∈ S(N). Then χ_𝐫 is the multiplier system of an eta-quotient of weight 0 and cover index 2. According to the third property in Remark <ref>, χ_𝐫 descends to a character on Γ_0(N). Thus 𝒮(N) can be regarded as a subgroup of the group of unitary linear characters of Γ_0(N).


The sets given in the following definition are subsets of 𝒟(N) that occur in the complete system of representatives of 𝒮(N).

Let N be a positive integer and p be a prime divisor of N. If p≠2,3 then define ℬ_N^p={p}. For p=2, we define

    ℬ_N^2=
    {2^1}    if v_2(N)=1,
    {2^1,2^2}    if v_2(N)=2,
    {2^α-2,2^α-1,2^α}    if v_2(N)=α≥3.

For p=3, we define

    ℬ_N^3=
    {3^1}    if v_3(N)=1,
    {3^α-1,3^α}    if v_3(N)=α≥2.

Finally, set ℬ_N=∪_p| Nℬ_N^p. Note that ℬ_1=∅.

We introduce a useful notation which will be used in the proofs below: if 𝐫_1,𝐫_2∈ S(N), then 𝐫_1∼_N𝐫_2, or simply 𝐫_1∼𝐫_2, means 𝐫_1-𝐫_2∈ Z(N). This is an equivalence relation compatible with the addition of S(N).


Let N=2^α_0· 3^α_1· N_1 with (N_1,6)=1 and α_0,α_1∈Z0.

  
  * For any 0≤β_0≤α_0, 0≤β_1≤α_1 and n_1,n_2| N_1 with (n_1,n_2)=1, there exist x,y,z≡1 2 such that
  
    e_2^β_03^β_1n_1n_2∼ xe_2^β_03^β_1n_1+ye_2^β_03^β_1n_2+ze_2^β_03^β_1.

  
  * For any 0≤β_0≤α_0, 0≤β_1≤α_1 and n| N_1, there exist x,y≡1 2 and z∈ such that
  
    e_2^β_03^β_1n∼ xe_2^β_03^β_1+ye_n+ze_1.

  
  * For any 0≤β_0≤α_0, 0≤β_1≤α_1 , there exist x,y,z∈ such that
  
    e_2^β_03^β_1∼ xe_2^β_0+ye_3^β_1+ze_1.




First we prove (<ref>). If n_1=1 or n_2=1 then (<ref>) is immediate. Thus assume that n_1,n_2>1. By the definition of Z(N), a solution (x,y,z)∈^3 of the system

    2^β_03^β_1n_1x+2^β_03^β_1n_2y+2^β_03^β_1z ≡ 2^β_03^β_1n_1n_2 24
    N/2^β_03^β_1n_1x+N/2^β_03^β_1n_2y+N/2^β_03^β_1z ≡N/2^β_03^β_1n_1n_224
    
    x,y,z≡ 1  2

is also a solution of (<ref>). Moreover, a solution of the system

    n_1x+n_2y+z ≡ n_1n_2 24
    
    n_2x+n_1y+n_1n_2z ≡ 1 24
    
    x,y,z≡ 1  2

is a solution of (<ref>). Since 2,3∤ n_1,n_2 we have n_1^2≡ 1 24 and n_2^2≡ 1 24. Thus the first equation of (<ref>) is equivalent to the second one. Therefore let x,y≡ 1  2 be arbitrary and set z≡ n_1n_2-n_1x-n_2y; we obtain a solution of (<ref>) and hence of (<ref>).

The proofs of (<ref>) and (<ref>) are similar of which we omit the details.




Any element in 𝒮(N) can be represented as

    ∑_n∈ℬ_Nc_ne_n-(∑_n∈ℬ_Nc_n)e_1+Z(N),

where c_n∈ and 0≤ c_n<24 for any n∈ℬ_N.

As a consequence, 𝒮(N) is a finite group and 𝒮(N)≤ 24^ℬ_N.

Let 𝐫+Z(N)=∑ r_ne_n+Z(N)∈𝒮(N) be arbitrary. According to Lemma <ref> we can assume that r_n=0 unless n is a prime power or n=1. Therefore there exist 𝐫^(p)+Z(N)=∑_0≤β≤ v_p(N)r^(p)_p^βe_p^β+Z(N)∈𝒮(N) such that 𝐫∼∑_p| N𝐫^(p). (Explicitly, let r^(p)_p^β=r_p^β for β>0 and r^(p)_p^0=-∑_β>0r_p^β.) It remains to find c_n∈ such that

    𝐫^(p)∼∑_n∈ℬ_N^pc_ne_n-(∑_n∈ℬ_N^pc_n)e_1

and 0≤ c_n < 24 for each p| N. If p≠2,3 then (<ref>) is equivalent to

    ∑_1≤β≤ v_p(N)(p^β-1)r^(p)_p^β≡ (p-1)c_p 24
    ∑_1≤β≤ v_p(N)(Np^-β-N)r^(p)_p^β≡ (Np^-1-N)c_p 24
    
    c_p≡∑_β≥1, r^(p)_p^β≡12β 2.

Since p^2≡ 124, the solution of this system is c_p≡∑_2∤βr^(p)_p^β24/(12,p-1). If p=2 and v_2(N)=α≥3, then (<ref>) is equivalent to

    ∑_1≤β≤α(2^β-1)r^(2)_2^β≡ (2^α-1)c_2^α+(2^α-1-1)c_2^α-1+(2^α-2-1)c_2^α-224
    ∑_1≤β≤α(N2^-β-N)r^(2)_2^β≡ (N2^-α-N)c_2^α+(N2^-α+1-N)c_2^α-1+(N2^-α+2-N)c_2^α-224
    ∑_β∈{α-2,α-1,α}c_2^β≡ r^(2)_2^β+12β≡∑_1≤β<α-2, r^(2)_2^β≡12β 2.

By considering the subcases 2|α≥6, 2∤α≥5 and α=3,4 separately and splitting the congruences modulo 24 to congruences modulo 3 and 8, one can verify that this system is solvable. We omit the proof of the case p=2, v_2(N)=1,2. If p=3 and v_3(N)=α≥2, then 
(<ref>) is equivalent to

    ∑_1≤β≤α(3^β-1)r^(3)_3^β≡ (3^α-1)c_3^α+(3^α-1-1)c_3^α-124
    ∑_1≤β≤α(N3^-β-N)r^(3)_3^β≡ (N3^-α-N)c_3^α+(N3^-α+1-N)c_3^α-124
    ∑_β∈{α-1,α}c_3^β≡ r^(3)_3^β+12β≡∑_1≤β<α-1, r^(3)_3^β≡12β 2.

As in the above case, by considering the subcases 2|α and 2∤α separately and splitting the congruences modulo 24 to congruences modulo 3 and 8, one can verify that this system is solvable. The case p=3, v_3(N)=1 is obvious. Thereby we have shown (<ref>) which concludes the proof.


To state the main theorem of this section, let us introduce the system of congruences

    ∑_n∈ℬ_N(n-1)c_n≡0 24
    ∑_n∈ℬ_N(Nn^-1-N)c_n≡0 24
    
    c_p≡02, ∀ p| N,p≠2,3
    ∑_2^β∈ℬ_N c_2^β≡12β≡02,  ∑_3^β∈ℬ_N c_3^β≡12β≡02.

Then the element (<ref>) is the zero in 𝒮(N) if and only if (c_n)_n∈ℬ_N satisfies (<ref>) according to the definition of Z(N).


Let N∈Z2. For any ordering b_1,b_2,…,b_t of ℬ_N (which means ℬ_N={b_1,b_2,…,b_t} and b_i≠ b_j for i≠ j), we define a sequence of positive integers Δ_1,Δ_2,…,Δ_t as follows:

    Δ_i=min{1≤ m≤ 24∃(c_n)_n∈ℬ_N s.t. (<ref>) holds and c_b_j=0 (j<i), c_b_i=m}.


Note that Δ_i is well defined since the set in the right-hand side always contains 24. In addition, be careful that the definition does not imply ∑_1≤ j≤ tΔ_je_j-(∑_1≤ j≤ tΔ_j)e_1∈ Z(N) unless ℬ_N=1.


With the notation of Definition <ref>, a complete system of representatives of 𝒮(N) is given by

    ∑_1≤ j≤ tc_b_je_b_j-(∑_1≤ j≤ tc_b_j)e_1,   0≤ c_b_j<Δ_j.



Let 𝐫+Z(N)∈𝒮(N) be arbitrary. According to Lemma <ref> we can assume that 𝐫=∑_n∈ℬ_Nr_ne_n-(∑_n∈ℬ_Nr_n)e_1 with r_n∈. By the definition of Δ_1, there exists 𝐝_1=∑_n∈ℬ_Nd_ne_n-(∑_n∈ℬ_Nd_n)e_1∈ Z(N) with d_b_1=Δ_1. Hence subtracting a multiple of 𝐝_1 from 𝐫 we may assume that 0≤ r_b_1<Δ_1. Inductively, suppose we have adjusted 𝐫 such that 0≤ r_b_j<Δ_j for j=1,2,…,J. Let 𝐝_J+1=∑_n∈ℬ_Nd_ne_n-(∑_n∈ℬ_Nd_n)e_1∈ Z(N) with d_b_j=0 (j≤ J) and d_b_J+1=Δ_J+1. This exists by the definition of Δ_J+1. Subtracting a multiple of 𝐝_J+1 from 𝐫 we can assume that 0≤ r_b_j<Δ_j for j=1,2,…,J+1. Therefore by induction 𝐫 takes the form of (<ref>) modulo Z(N). We have proved that each coset in 𝒮(N) can be represented by an element in (<ref>).

Let 𝐫_1=∑_1≤ j≤ tc_b_je_b_j-(∑_1≤ j≤ tc_b_j)e_1 and 𝐫_2=∑_1≤ j≤ td_b_je_b_j-(∑_1≤ j≤ td_b_j)e_1 be two different representatives of the form (<ref>). Let 1≤ J ≤ t satisfy c_b_j=d_b_j for j<J and c_b_J≠ d_b_J. Without loss of generality assume that c_b_J<d_b_J. It follows that 0<d_b_J-c_b_J<Δ_J. Then 𝐫_2-𝐫_1∉Z(N) by the definition of Δ_J. We have proved that the vectors in (<ref>) are pairwise inequivalent modulo Z(N), which concludes the whole proof.



Let N∈Z1 and k∈1/2. Then the characters (<ref>) of eta-quotients of level N, weight k and cover index 2 are in one-to-one correspondence with the vectors

    ∑_1≤ j≤ tr_b_je_b_j+(2k-∑_1≤ j≤ tr_b_j)e_1,   0≤ r_b_j<Δ_j.



According to the discussion followed by Remark <ref>, the characters of eta-quotients of level N, weight k and cover index 2 can be regarded as the set 𝒮(N,k) and the map 𝒮(N)→𝒮(N,k), 𝐫+Z(N)↦𝐫+2ke_1+Z(N) is a bijection. The corollary follows from this and Theorem <ref>.


In the remainder of this section, we give examples of concrete sequences (Δ_i)_1≤ i≤ t (t=ℬ_N) for some special N.


Let N=p^α where p≥ 5 is a prime and α∈Z1. Then ℬ_N={p} and the sequence (Δ_i)_1≤ i≤ t (t=ℬ_N) contains a single element Δ_1. The system of congruences (<ref>) is equivalent to (p-1)c_p≡024 and c_p≡02 in this case. Thus, Δ_1=24/(12,p-1) and a complete system of representatives of 𝒮(p^α) is given by

    r_p(e_p-e_1),   0≤ r_p<24/(12,p-1).

As a consequence, for any k∈1/2 there are exactly 24/(12,p-1) characters χΓ_0(p^α)→^× of eta-quotients of level p^α, weight k and cover index 2. It should be noted that, the character of an eta-quotient, say, of cover index 4 may also be regarded as a character on Γ_0(p^α) (cf. the third fact of Remark <ref>). For instance, the character χΓ_0(p^α)8→^× of η(τ)^1/4η(pτ)^3/4 descends to a character on Γ_0(p^α)=Γ_0(p^α)2. Such characters are not in the list above.




Let N=2^α with α≥3. Then ℬ_N={2^α-2,2^α-1,2^α} which is arranged in the increasing order. We need to calculate the sequence Δ_1,Δ_2,Δ_3. The system of congruences (<ref>) is equivalent to

    (2^α-2-1)c_2^α-2+(2^α-1-1)c_2^α-1+(2^α-1)c_2^α≡024
    
    (4-2^α)c_2^α-2+(2-2^α)c_2^α-1+(1-2^α)c_2^α≡024
    ∑_β∈{α-2,α-1,α}c_2^β≡1 2β≡02.

One can verify directly that

    (Δ_1,Δ_2,Δ_3)=
    (2,24,8)      if α≥3 and 2|α
    
    (2,8,24)      if α≥3 and 2∤α.

Therefore, a complete system of representatives of 𝒮(2^α) with α≥3 is given by

    r_2^α-2e_2^α-2+r_2^α-1e_2^α-1+r_2^αe_2^α-(r_2^α-2+r_2^α-1+r_2^α)e_1,

where r_α-2∈{0,1} and 0≤ r_2^α-1<24, 0≤ r_2^α<8 if 2|α; 0≤ r_2^α-1<8, 0≤ r_2^α<24 if 2∤α. As a consequence, for any k∈1/2 there are exactly 384 characters χΓ_0(2^α)→^× of eta-quotients of level 2^α, weight k and cover index 2 when α≥3. In addition, it is immediate that for N=2^2 we have ℬ_4={2^1,2^2} and (Δ_1,Δ_2)=(24,8) and for N=2^1 we have ℬ_2={2^1} and Δ_1=24. Thus, there are exactly 192 (24 respectively) characters of eta-quotients of level 4 (2 respectively), cover index 2 and any fixed weight ∈1/2.




Let N=3^α with α≥2. Then ℬ_N={3^α-1,3^α} which is arranged in the increasing order. We need to calculate the sequence Δ_1,Δ_2. The system of congruences (<ref>) is equivalent to

    (3^α-1-1)c_3^α-1+(3^α-1)c_3^α≡024
    
    (3-3^α)c_3^α-1+(1-3^α)c_3^α≡024
    ∑_β∈{α-1,α}c_3^β≡1 2β≡02.

One can verify directly that

    (Δ_1,Δ_2)=
    (12,3)      if α≥2 and 2|α
    
    (3,12)      if α≥2 and 2∤α.

Therefore, a complete system of representatives of 𝒮(3^α) with α≥2 is given by

    r_3^α-1e_3^α-1+r_3^αe_3^α-(r_3^α-1+r_3^α)e_1,

where 0≤ r_3^α-1<12, 0≤ r_3^α<3 if 2|α; 0≤ r_3^α-1<3, 0≤ r_3^α<12 if 2∤α. As a consequence, for any k∈1/2 there are exactly 36 characters χΓ_0(3^α)→^× of eta-quotients of level 3^α, weight k and cover index 2 when α≥2. In addition, it is direct that for N=3^1 we have ℬ_3={3^1} and Δ_1=12. Thus, there are exactly 12 characters of eta-quotients of level 3, cover index 2 and any fixed weight ∈1/2.




Let N=4p^α where p≥5 is a prime and α∈Z1. Then ℬ_N={2^1,2^2,p} which is arranged in the increasing order. We need to calculate the sequence Δ_1,Δ_2,Δ_3. The system of congruences (<ref>) is equivalent to

    c_2+3c_4+(p-1)c_p≡024
    
    -2p^α c_2-3p^α c_4+(4p^α-1-4p^α)c_p≡024
    
    c_2, c_p≡02.

Using elementary number theory one can show that

    Δ_1=2·(12,p-1),  Δ_2=8,  Δ_3=24/(12,p-1).

Therefore, a complete system of representatives of 𝒮(4p^α) with p≥5 and α≥1 is given by

    r_2e_2+r_4e_4+r_pe_p-(r_2+r_4+r_p)e_1

with

    0≤ r_2<2·(12,p-1),   0≤ r_4<8,   0≤ r_p<24/(12,p-1).

As a consequence, for any k∈1/2 there are exactly 384 characters χΓ_0(4p^α)→^× of eta-quotients of level 4p^α, weight k and cover index 2 when p≥5 and α≥1.




Let N=p_1^α_1p_2^α_2 where p_1,p_2≥5 are distinct primes and α_1,α_2∈Z1. Then ℬ_N={p_1,p_2}. We need to calculate the sequence Δ_1,Δ_2. The system of congruences (<ref>) is equivalent to

    (p_1-1)c_p_1+(p_2-1)c_p_2≡024
    
    (p_1^α_1-1p_2^α_2-p_1^α_1p_2^α_2)c_p_1+(p_1^α_1p_2^α_2-1-p_1^α_1p_2^α_2)c_p_2≡024
    
    c_p_1, c_p_2≡02.

Using elementary number theory one can show that

    Δ_1   =24·(12,p_2-1)/(12,(p_1-1)(p_2-1))·(12·(12,p_2-1)/(12,(p_1-1)(p_2-1)),p_1-p_2)^-1,
    Δ_2   =24/(12,p_2-1).

The expression for Δ_1 seems to be complicated but it actually depends only on p_112 and p_212. For instance, if p_2≡112, then Δ_1 is simplified to 24/(12,p_1-1). A complete system of representatives of 𝒮(p_1^α_1p_2^α_2) is given by

    r_p_1e_p_1+r_p_2e_p_2-(r_p_1+r_p_2)e_1

with

    0≤ r_p_1<Δ_1,   0≤ r_p_2<Δ_2.

As a consequence, for any k∈1/2 there are exactly Δ_1Δ_2 characters χΓ_0(p_1^α_1p_2^α_2)→^× of eta-quotients of level p_1^α_1p_2^α_2, weight k and cover index 2 when p_1, p_2≥5 are distinct primes and α_1,α_2≥1.





§ DIMENSION FORMULAS FOR WEIGHTS 1/2, 1, 3/2 AND SMALL LEVELS


In this section, we give values of all _ M_k(Γ_0(N),χ) that can be calculated by Theorem <ref> (throughout this section, t in this theorem always takes the value 0) for k=1/2, 1 and 3/2, N∈Z1 and χ being the character of an eta-quotient of level N, weight k and cover index 2. There are only finitely many such formulas since the right-hand side of (<ref>) tends to 2 when N→+∞ and the set of χ is finite for each N and k (cf. Corollary <ref>). The following lemma gives an upper bound of N that should be considered.


Let N be a positive integer and let m, ε_2, ε_3 and ε_∞ be defined in (<ref>), (<ref>), (<ref>) and (<ref>) respectively. If N≥1024, we have

    ε_2<m/96,  ε_3<m/128,  ε_∞≤m/32.

Consequently, in Theorem <ref>, if k≤3/2, t=0 and N≥1024, then (<ref>) does not hold.

For ε_∞, a stronger estimate[This is the best estimate if we want it to hold for all N since it becomes equality when N is a square.] is ε_∞≤ m· N^-1/2 with N∈Z1 which we will prove below.

Suppose N=∏_1≤ j≤ sp_j^α_j where p_1<p_2<…<p_s are primes, s∈Z0 and α_j∈Z1. Note that for p=p_j, α=α_j we have

    p^[α/2]+p^[α/2-1/2]/p^α+p^α-1=2/p^α/2(p^1/2+p^-1/2)    if  2∤α,
    1/p^α/2    if  2|α.

Thereby, according to <cit.>

    ε_∞/m=∏_1≤ j≤ sp_j^[α_j/2]+p_j^[α_j/2-1/2]/p_j^α_j+p_j^α_j-1≤∏_1≤ j ≤ s1/p_j^α_j/2=N^-1/2.


Now we begin to prove the inequalities for ε_2 and ε_3, so suppose N≥1024. If s=1,2,3, then ε_2/m≤2^3/1025<1/96, ε_3/m≤2^3/1025<1/128. If s≥4, we consider two subcases: ε_2=0 and ε_2≠0. If ε_2=0, obviously ε_2/m<1/96. If ε_2≠0, then -4p_j≠-1 and hence p_j=2 or p_j≡1 4 for 1≤ j≤ s. If p_1≠2, then p_1^α_1≥ 5, p_2^α_2≥ 13 and p_j^α_j≥ 17 for j≥3. Therefore

    ε_2/m≤2/5+1·2/13+1·(2/17+1)^s-2<1/96.

If p_1=2, then p_2^α_2≥ 5, p_3^α_3≥ 13 and p_j^α_j≥ 17 for j≥4. Therefore

    ε_2/m≤1/2+1·2/5+1·2/13+1·(2/17+1)^s-3<1/96.

This proves the inequality for ε_2. To prove the inequality for ε_3 when s≥4, we consider the subcases ε_3=0 and ε_3≠0 separately and proceed as in the proof for ε_2.

Finally, with the notation of Theorem <ref> suppose that k≤3/2, t=0 and N≥1024. Assume by contradiction that (<ref>) holds, then

    3/2≥ k > 2-6/mε_2-8/mε_3-12/mε_∞>2-6/96-8/128-12/32

which is absurd. Hence (<ref>) does not hold.


The algorithm for presenting all _ M_k(Γ_0(N),χ) described above is as follows:

  
  * Let k range over {1/2,1,3/2} and let N range over the positive integers ≤1023. We do not miss anything according to Lemma <ref>.
  
  * Given k,N, we arrange ℬ_N in any order and calculate the sequence Δ_1,…,Δ_t (cf. Definition <ref>).
  
  * Let 𝐫=(r_n)_n| N range over the vectors described in Corollary <ref>. According to this corollary, the corresponding χ_𝐫 ranges over exactly all characters of eta-quotients of level N, weight k and cover index 2.
  
  * Given k,N and 𝐫, we calculate 𝐱=(x_c)_c| N via (<ref>).
  
  * We determine whether (<ref>) holds. If it holds, then we calculate _ M_k(Γ_0(N),χ_𝐫) by (<ref>) where t=0. If it does not hold, _ M_k(Γ_0(N),χ_𝐫) can not be calculated by Theorem <ref>. Nevertheless we can obtain an upper bound and a lower bound of _ M_k(Γ_0(N),χ_𝐫) (cf. Remark <ref>).



For a fixed k, one can shrink the range of N by picking out those N≤1023 such that k>2-6/mε_2-8/mε_3-12/mε_∞. Hence, for k=1/2, N need only to range over {1≤ N≤21}∪{24,25,27,32,36,49,50}; for k=1, N need only to range over {1≤ N ≤ 22}∪{24≤ N ≤ 32}∪{34,36,37,39,40,45,48,49,50,54,64,72,75,81,98,100,121,169}; for k=3/2, N need only to range over a subset of {1≤ N ≤529}.

All the above steps can be easily implemented in any computer algebra system. We program them using SageMath. See Appendix <ref> for the usage of the code.



 §.§ The weight 1/2


The dimensions obtained by the above algorithm are summarized in Table <ref>.


Let us consider the case N=50. We have ℬ_50={2,5} and (Δ_1,Δ_2)=(8,6) as one can verify directly. Thus, there are totally 48 characters of eta-quotients of level 50, weight 1/2 and cover index 2. They are exactly the characters on Γ_0(50)2 of the following eta-quotients:

    η(τ)^1-r_2-r_5η(2τ)^r_2η(5τ)^r_5,   0≤ r_2<8, 0≤ r_5<6.

Among the 48 characters, for precisely the four characters χ given by

    (r_2,r_5)=(0,0), (1,0), (2,0), (7,4)

one can compute _ M_1/2(Γ_0(50),χ) using Theorem <ref> (that is, (<ref>) holds with t=0) and the four dimensions all equal 2. For the other 44 characters χ our method only gives that 0≤_ M_1/2(Γ_0(50),χ)≤4 (cf. Remark <ref>). Let χ_1,χ_2,χ_3,χ_4 denote respectively the characters given by (<ref>). Then according to (<ref>) and Lemma <ref> we have

    M_1/2(Γ_0(50),χ_1)   =η(τ)⊕η(25τ),
    
    M_1/2(Γ_0(50),χ_2)   =η(2τ)⊕η(50τ),
    
    M_1/2(Γ_0(50),χ_3)   =η(τ)^-1η(2τ)^2⊕η(25τ)^-1η(50τ)^2.

Finding explicit generators of M_1/2(Γ_0(50),χ_4) seems to be a bit harder since η(τ)^-10η(2τ)^7η(5τ)^4 is only a weakly holomorphic modular form. However, we have 7e_2+4e_5-11e_1∼-e_2+e_1 module Z(50) which can be verified by (<ref>). It follows that

    M_1/2(Γ_0(50),χ_4)=η(τ)^2η(2τ)^-1⊕η(25τ)^2η(50τ)^-1.

Finally note that M_1/2(Γ_0(50),χ_j)=S_1/2(Γ_0(50),χ_j) for j=1,2, while the Eisenstein subspace of M_1/2(Γ_0(50),χ_j) is nontrivial for j=3,4.



We consider another example N=20 in which there are totally 384 characters. They have been described in Example <ref> with p=5, α=1. We have ℬ_20={2,4,5} and (Δ_1,Δ_2,Δ_3)=(8,8,6). Therefore the characters are exactly those on Γ_0(20)2 of the following eta-quotients:

    η(τ)^1-r_2-r_4-r_5η(2τ)^r_2η(4τ)^r_4η(5τ)^r_5,   0≤ r_2<8, 0≤ r_4<8, 0≤ r_5<6.


Among the 384 characters, for precisely the six characters χ given by

    (r_2,r_4,r_5)=(1,6,1), (3,0,5), (3,2,5), (5,6,0), (7,0,4), (7,2,4)

one can compute _ M_1/2(Γ_0(20),χ) using Theorem <ref> and the six dimensions all equal 1. For the other 378 characters χ our method only gives that 0≤_ M_1/2(Γ_0(20),χ)≤2 (cf. Remark <ref>). Let χ_1,χ_2,χ_3,χ_4,χ_5,χ_6 denote respectively the characters given by (<ref>). One can verify using (<ref>) and Lemma <ref> that

    M_1/2(Γ_0(20),χ_1)   =η(5τ)^-2η(10τ)^5η(20τ)^-2,     M_1/2(Γ_0(20),χ_2)   =η(5τ)^2η(10τ)^-1,
    
    M_1/2(Γ_0(20),χ_3)   =η(10τ)^-1η(20τ)^2,     M_1/2(Γ_0(20),χ_4)   =η(τ)^-2η(2τ)^5η(4τ)^-2,
    
    M_1/2(Γ_0(20),χ_5)   =η(τ)^2η(2τ)^-1,     M_1/2(Γ_0(20),χ_6)   =η(2τ)^-1η(4τ)^2.



There are totally 188 nonzero spaces M_1/2(Γ_0(N),χ) whose dimensions can be computed by Theorem <ref> according to Table <ref>. Note that we restrict ourselves to the characters χ of eta-quotients of cover index 2, so the dimension formulas summarized in Table <ref> are not direct consequences of the well known theorem of Serre and Stark <cit.> that gives an explicit basis of M_1/2(Γ_0(N),χ_1χ) where 4 | N, χ_1 is the character of Euler theta function θ(τ)=η(τ)^-2η(2τ)^5η(4τ)^-2 and χ is the character induced by any even Dirichlet character (/N)^×→^×. For instance, when N=12, there are two even Dirichlet characters and hence the theorem of Serre and Stark gives explicit bases of two spaces M_1/2(Γ_0(12),χ). As a comparison, there are totally 1152 characters of eta-quotients of weight 1/2, level 12 and cover index 2 among which the dimensions of 318 spaces can be computed by Theorem <ref>—286 spaces are zero-dimensional and 32 spaces are one-dimensional.



Zagier observed that there are exactly 14 primitive (which means the greatest common divisor of all the n with r_n≠ 0 is equal to 1) eta-quotients of integral exponents that are modular forms of weight 1/2. This and the similar conjectures on higher weights were proved by Mersmann in his Master's thesis. See <cit.> for a simpler proof. It seems that all the generators of the 188 nonzero spaces M_1/2(Γ_0(N),χ) whose dimensions can be computed by Theorem <ref> (cf. Table <ref>) take the form f(mτ) where f is one of the 14 primitive eta-quotients and m is a suitable positive integer. This is verified by the SageMath program for N=4,20,50. See Appendix <ref> for how to find the code doing this.









 §.§ The weight 1


The dimensions obtained by the algorithm described above are summarized in Table <ref>. See Appendix <ref> for the SageMath code that generates this kind of tables. Note that the characters are originally defined on Γ_0(N)2 since we are considering eta-quotients of cover index 2 but they actually descend to characters on the matrix groups Γ_0(N) since the weight is 1.
llllllllllll
For each N, a means the total number of characters (cf. Corollary <ref>) and v means the number of characters χ such that _ M_1(Γ_0(N),χ) can be computed using Theorem <ref>, d_j means the number of spaces of dimension j for 0≤ j ≤ 8. 

N     a     v     d_0     d_1     d_2     d_3     d_4     d_5     d_6     d_7     d_8 

N     a     v     d_0     d_1     d_2     d_3     d_4     d_5     d_6     d_7     d_8 

1     1     1     0     1     0     0     0     0     0     0     0

2     24     24     17     7     0     0     0     0     0     0     0

3     12     12     7     5     0     0     0     0     0     0     0

4     192     173     142     31     0     0     0     0     0     0     0

5     6     6     3     3     0     0     0     0     0     0     0

6     288     250     189     60     1     0     0     0     0     0     0

7     4     4     1     3     0     0     0     0     0     0     0

8     384     331     247     83     1     0     0     0     0     0     0

9     36     29     17     11     1     0     0     0     0     0     0

10     48     46     20     22     4     0     0     0     0     0     0

11     12     1     0     1     0     0     0     0     0     0     0

12     1152     947     617     305     24     1     0     0     0     0     0

13     2     2     0     1     1     0     0     0     0     0     0

14     96     21     0     20     1     0     0     0     0     0     0

15     72     18     0     16     2     0     0     0     0     0     0

16     384     293     166     108     18     1     0     0     0     0     0

17     6     4     0     4     0     0     0     0     0     0     0

18     288     217     113     75     25     3     1     0     0     0     0

19     4     3     0     3     0     0     0     0     0     0     0

20     384     96     0     72     24     0     0     0     0     0     0

21     24     14     0     8     6     0     0     0     0     0     0

22     96     1     0     0     1     0     0     0     0     0     0

24     2304     658     0     480     176     0     2     0     0     0     0

25     6     4     1     1     1     1     0     0     0     0     0

26     48     4     0     0     4     0     0     0     0     0     0

27     36     15     0     9     5     1     0     0     0     0     0

28     384     11     0     0     10     1     0     0     0     0     0

30     576     2     0     0     0     0     2     0     0     0     0

32     384     146     0     100     36     8     2     0     0     0     0

36     1152     412     0     198     168     30     15     0     1     0     0

37     2     1     0     0     1     0     0     0     0     0     0

40     768     2     0     0     0     0     2     0     0     0     0

45     72     2     0     0     0     0     2     0     0     0     0

48     2304     58     0     0     0     16     40     0     2     0     0

49     4     2     0     0     1     0     1     0     0     0     0

50     48     26     0     0     19     0     0     7     0     0     0

54     288     1     0     0     0     0     0     0     1     0     0

64     384     18     0     0     0     0     16     0     2     0     0

72     2304     2     0     0     0     0     0     0     0     0     2

75     72     2     0     0     0     0     0     0     2     0     0

81     36     1     0     0     0     0     0     0     1     0     0

98     96     1     0     0     0     0     0     0     0     0     1

121     12     1     0     0     0     0     0     0     1     0     0



We focus on the level 98 in the rest of this subsection. According to Table <ref>, there are totally 96 characters χ of eta-quotients of weight 1, level 98 and cover index 2. They correspond to 96 independent spaces M_1(Γ_0(98),χ) among which the dimension of exactly one space can be computed by Theorem <ref>, namely, the space where χ is the character of η(τ)^-15η(2τ)^16η(7τ). For this χ we have _ M_1(Γ_0(98),χ)=8. One can show that χabcd=-7d by, for instance, a result of Gordon, Hughes and Newman (cf. <cit.>) or by checking directly for abcd being in a set of generators of Γ_0(98). Thus, an explicit basis can be given using Eisenstein series constructed by Weisinger <cit.>. See also <cit.>. This shows there is no cusp form in M_1(Γ_0(98),χ), which can also be proved using the tools developed in <cit.>.

One may ask: is the space M_1(Γ_0(98),χ) generated by holomorphic eta-quotients? The answer is no. According to (<ref>), the functions

    η(τ)^-1η(2τ)^2η(7τ)^-1η(14τ)^2   =q+q^2+q^4+q^7+q^8+q^9+2q^11+q^14+…
    η(τ)^2η(2τ)^-1η(7τ)^2η(14τ)^-1   =1-2q+2q^4-2q^7+4q^8-2q^9-4q^11+6q^16+…
    η(7τ)^-1η(14τ)^2η(49τ)^-1η(98τ)^2   =q^7+q^14+q^28+q^49+q^56+q^63+2q^77+q^98+…
    η(7τ)^2η(14τ)^-1η(49τ)^2η(98τ)^-1   =1-2q^7+2q^28-2q^49+4q^56-2q^63-4q^77+6q^112+…

are holomorphic eta-quotients of level 98. Using Lemma <ref> the characters of these four modular forms are equal to χ and hence they belong to M_1(Γ_0(98),χ). By their q-coefficients these four functions are linearly independent. However there are no other holomorphic eta-quotients of cover index 2 in M_1(Γ_0(98),χ) so this space can not be generated by a set consisting of all eta-quotients. This can be seen as follows. Let f(τ)=∏_n|98η(nτ)^r_n be in M_1(Γ_0(98),χ) with r_n∈. Then x_c∈Z0 for any c| 98 where x_c is given by (<ref>). According to Theorem <ref> (applied to f), (<ref>) and the complete set of representatives of Γ_0(98)\ described in the proof of Proposition <ref> we have x_1+x_2+6x_7+6x_14+x_49+x_98=336 of which there are only finitely many solutions (x_c)_c| 98. For each solution, we obtain (r_n)_n|98 via (<ref>) and then check whether r_n∈ and the character corresponding to (r_n)_n|98 is the same as χ using Lemma <ref>. In this way we find that there are exactly four holomorphic eta-quotients of cover index 2 in M_1(Γ_0(98),χ) (totally 69 holomorphic eta-quotients, 57 of which are primitive).





































 §.§ The weight 3/2


There are totally 17862 spaces M_3/2(Γ_0(N),χ) whose dimensions can be calculated using Theorem <ref>. Among these spaces, the largest level is N=400 and the largest dimension is 48. See Appendix <ref> for the usage of SageMath code that gives these data. The following observation will be used later: the dimensions of these 17862 spaces are all greater than 1 when N>36.




§ APPLICATION: ONE-DIMENSIONAL SPACES AND EISENSTEIN SERIES OF RATIONAL WEIGHTS


In <cit.>, the author gives infinitely many identities whose left-hand sides are Eisenstein series and right-hand sides are eta-quotients of rational weights for the levels 2 and 3. Using Theorem <ref> (with t=0), we find new identities of this kind for the level 4 in this section.

First let us rewrite Theorem <ref> more explicitly for N=4. We have m=6, ε_2=ε_3=0. Let r_1, r_2, r_4 be rational numbers and set k=1/2(r_1+r_2+r_4). Let

    [ x_1; x_2; x_4 ]=
    [ 4 2 1; 1 2 1; 1 2 4 ]·[ r_1; r_2; r_4 ].

Our aim is to find an Eisenstein series E(τ) of weight k equal to f(τ)=η(τ)^r_1η(2τ)^r_2η(4τ)^r_4. We only consider the Eisenstein series defined at the cusp ∞, so assume x_4=0 which means f does not vanish at the cusp ∞. Since f must be a holomorphic eta-quotient, we assume x_1,x_2≥0. Therefore,

    x_1=-6r_2-15r_4≥0,   x_2=-3r_4≥0,  hence k=1/2(-r_2-3r_4)≥0.

Now the first inequality in (<ref>) becomes k>-4+2({x_1/24}+{x_2/24}), which holds since k≥0. Consequently, (<ref>) with t=0 is applicable. We summarize:


Let r_2 and r_4 be rational numbers satisfying -2r_2-5r_4≥0 and -r_4≥0. Let D∈Z1 such that Dr_2, Dr_4∈2. Then we have

    _ M_k(Γ_0(4), χ)=[-2r_2-5r_4/8]+[-r_4/8]+1,

where χΓ_0(4)D→^× is the character of η(τ)^-2r_2-4r_4η(2τ)^r_2η(4τ)^r_4 and k=1/2(-r_2-3r_4).


We have shown that (<ref>) with t=0 holds. A simplification gives the desired formula.



Let the notation and assumptions be as in Proposition <ref>. Then _ M_k(Γ_0(4), χ)=1 if and only if

    0≤-2r_2-5r_4<8  and  0≤-r_4<8.

Moreover, suppose _ M_k(Γ_0(4), χ)=1; then k>2 if and only if (-2r_2-5r_4)+(-r_4)>8.

Note that when _ M_k(Γ_0(4), χ)=1, then M_k(Γ_0(4), χ)=η(τ)^-2r_2-4r_4η(2τ)^r_2η(4τ)^r_4. Hence if M_k(Γ_0(4), χ) contains an Eisenstein series, then an identity that relates the eta-quotient and the Eisenstein series follows. In the rest we only consider the case k>2 in which nonholomorphic Eisenstein series are avoided. Similar identities can also be derived in the cases k<2, which we will do in the future work.

Recall the definition of Eisenstein series of rational weights greater than 2:
[Definition 3.1, <cit.>]

Let D,N be positive integers, and let k∈1/D that is greater than 2. Let χΓ_0(N)D→^× be a finite index character such that χ(-I)=-k/2. Let s ∈ and γ_s ∈ such that γ_s(∞)=s. If χ(γ_sT^w_sγ_s^-1)=1, then we define the Eisenstein series E_γ_s,k on the group Γ_0(N)D, of weight k, with character χ and at cusp γ_s as

    E_γ_s,k(τ)=∑_γ∈γ_s⟨T^w_s⟩γ_s^-1\Γ_0(N)Dχ(γ)^-1· 1|_kγ_s^-1γ.


In the above definition, w_s is the width (c.f. Section <ref>). If s=a/c where a,c are coprime integers, then w_s=N/(N,c^2). The basic properties of these E_γ_s,k are collected in <cit.>.

Now let N=4 and let χ be as in Proposition <ref>. Since χ(T)=1 the Eisenstein series E_I,k(τ) is well-defined provided that k>2, that is, (-2r_2-5r_4)+(-r_4)>8. The main result of this section is the following:


Let r_2 and r_4 be rational numbers satisfying

    0≤-2r_2-5r_4<8,  0≤-r_4<8,  (-2r_2-5r_4)+(-r_4)>8.

Let D be a positive integer such that Dr_2, Dr_4∈2. Let χΓ_0(4)D→^× be the character of η(τ)^-2r_2-4r_4η(2τ)^r_2η(4τ)^r_4. Set k=1/2(-r_2-3r_4) and

    E_I,k(τ)=E_I,k(τ;r_2,r_4)=∑_γ∈⟨T⟩\Γ_0(4)Dχ(γ)^-1· 1|_kγ.

Then we have

    η(τ)^-2r_2-4r_4η(2τ)^r_2η(4τ)^r_4=1/2DE_I,k(τ;r_2,r_4).



Corollary <ref> shows that _ M_k(Γ_0(4), χ)=1. Let x_1, x_2 and x_4 be as in (<ref>) with r_1=-2r_2-4r_4. Then the assumptions imply that x_1≥0, x_2≥0 and x_4=0. Thus η(τ)^-2r_2-4r_4η(2τ)^r_2η(4τ)^r_4 is holomorphic at all cusps and hence is in M_k(Γ_0(4), χ). By <cit.> we have E_I,k∈ M_k(Γ_0(4), χ). It follows that η(τ)^-2r_2-4r_4η(2τ)^r_2η(4τ)^r_4=c· E_I,k. Let τ→+∞; we find that c=1/2D again by <cit.>.


In the rest of this section we will write out the Fourier expansion of 1/2DE_I,k(τ) which will give us a more explicit form of the identity (<ref>).

For any rational numbers r_2,r_4 with k=1/2(-r_2-3r_4)>2 we have

    1/2DE_I,k(τ;r_2,r_4)=1+∑_4| c>0,d∈(c,d)=1-1/24P(c,d;r_2,r_4)·(cτ+d)^-k

where

    P(c,d;r_2,r_4)   =r_2·(Ψa2bc/2d-2Ψabcd)+r_4·(Ψa4bc/4d-4Ψabcd)
       =r_2·(12s(-d,c/2)-24s(-d,c)+3)+r_4·(12s(-d,c/4)-48s(-d,c)+9)

with a,b being any integers such that ad-bc=1.


According to <cit.>,

    E_I,k(τ;r_2,r_4)=D·∑_4| c∈,d∈(c,d)=1χabcd^-1·(cτ+d)^-k

where a,b are any integers such that ad-bc=1. By Remark <ref>, χ(-I)=-k/2. It follows that

    χabcd^-1·(cτ+d)^-k=χ-a-b-c-d^-1·(-cτ-d)^-k.

Thus

    1/2DE_I,k(τ;r_2,r_4)=1+∑_4| c>0,d∈(c,d)=1χabcd^-1·(cτ+d)^-k.

Substituting (<ref>) and (<ref>) in the above expression gives the desired formula.



With the notation being as above, we have

    1/2DE_I,k(τ;r_2,r_4)=1+^- k/2(2)^k/Γ(k)∑_n ∈Z1n^k-1a(n)· q^n,

where

    a(n)=∑_c∈Z11/(4c)^k∑_0≤ d<4c(d,4c)=1dn/4c-1/24P(4c,d;r_2,r_4).



For d∈, c∈Z1 set

    f(d,c)=-1/24P(c,d;r_2,r_4)    if 4| c  and (d,c)=1;
    
    0     if 4∤ c  and (d,c)=1;
    
    f(d/(d,c),c/(d,c))     if  (d,c)>1.

Thus, 1/2DE_I,k(τ;r_2,r_4)=1+ζ(k)^-1·∑_c∈Z1,d∈f(d,c)(cτ+d)^-k since ∑_t | (c,d)μ(t) = 0 if (c,d) > 1 and ∑_tμ(t)/t^k=ζ(k)^-1, where ζ is the Riemann zeta function and μ is the Möbius function. Note that f(d+c,c)=f(d,c). This fact, together with the Lipschitz summation formula

    ∑_n ∈1/(τ+n)^s=^- s/2(2)^s/Γ(s)∑_n ≥ 1n^s-1^2 nτ,  τ∈, (s)>1

implies that

    ∑_d ∈f(d,c)(cτ+d)^-k   = c^-k∑_0 ≤ d_1 < cf(d_1,c)∑_d_0 ∈(τ+d_1/c+d_0)^-k
       =^- k/2(2)^k/Γ(k)c^-k∑_n ∈Z1n^k-1∑_0 ≤ d < cf(d,c)dn/cq^n.

Therefore,

    1/2DE_I,k(τ;r_2,r_4)=1+^- k/2(2)^k/Γ(k)ζ(k)∑_n ∈Z1n^k-1∑_c ∈Z11/c^k∑_0 ≤ d < cf(d,c)dn/cq^n.

By the definition of f(d,c) we find that

    ∑_c ∈Z11/c^k∑_0 ≤ d < cf(d,c)dn/c   =∑_t∈Z1∑_c∈Z1,0≤ d<c(d,c)=t1/(tc)^kf(td,tc)dn/c
       =ζ(k)·∑_c∈Z1,0≤ d<4c(d,4c)=11/(4c)^kf(d,4c)dn/4c
       =ζ(k)·∑_c∈Z1,0≤ d<4c(d,4c)=11/(4c)^kdn/4c-1/24P(4c,d;r_2,r_4).

Inserting this into (<ref>) gives the desired formula.


The reader may compare the above proposition with <cit.>.


We can now give an equivalent statement of Theorem <ref> in a form that gives Fourier coefficients of certain infinite q-products:


Let r_2 and r_4 be rational numbers satisfying (<ref>). Let D be a positive integer such that Dr_2, Dr_4∈2. Let ∑_n∈Z0c(n)q^n be the holomorphic branch of the D-th root of the infinite product

    ∏_n∈Z1(1-q^n)^-2Dr_2-4Dr_4∏_n∈Z1(1-q^2n)^Dr_2∏_n∈Z1(1-q^4n)^Dr_4

with c(0)=1. Then for n≥ 1 we have

    c(n)=^- k/2(2)^k/Γ(k)n^k-1∑_c∈Z11/(4c)^k∑_0≤ d<4c(d,4c)=1dn/4c-1/24P(4c,d;r_2,r_4).

(See (<ref>) for the definition of P(4c,d;r_2,r_4).)


There are exactly D holomorphic branches and the one with the leading term 1 is the eta-quotient η(τ)^-2r_2-4r_4η(2τ)^r_2η(4τ)^r_4. Now the assertion follows from Theorem <ref> and Proposition <ref>.




Let us consider the special case r_2=0, r_4=-3/2 for which (<ref>) holds. Note that k=1/2(-r_2-3r_4)=9/4. Set D=4. Then Corollary <ref> shows

    ∏_n∈Z1(1-q^n)^6/√(∏_n∈Z1(1-q^4n)^3)=1+^-9 /8(2)^9/4/Γ(9/4)∑_n∈Z1n^5/4(∑_c∈Z11/(4c)^9/4∑_0≤ d<4c(d,4c)=1dn/4c-1/24P(4c,d;0,-3/2))· q^n,

where the square root is the principal branch √(∏_n∈Z1(1-q^4n)^3)=1+o(1) and

    P(4c,d;0,-3/2)=-18s(-d,c)+72s(-d,4c)-27/2.

(See (<ref>) for the definition of s(-d,c).) For numerical check (of this and the next examples) using SageMath programs, see Appendix <ref>.



Corollary <ref> contains as well some identities involving ordinary eta-quotients, that is, eta-quotients with integral exponents. For instance, set r_2=7, r_4=-4 and hence k=1/2(-r_2-3r_4)=5/2 and D=2. For this setting, (<ref>) holds. The identity reads

    ∏_n∈Z1(1-q^n)^2(1-q^2n)^7(1-q^4n)^-4
    
    =1+^-5/4(2)^5/2/Γ(5/2)∑_n∈Z1n^3/2∑_c∈Z11/(4c)^5/2∑_0≤ d<4c(d,4c)=1dn/4c-1/24P(4c,d;7,-4)· q^n

where

    P(4c,d;7,-4)=-48s(-d,c)+84s(-d,2c)+24s(-d,4c)-15.

We will return to this function, i.e., η(τ)^2η(2τ)^7η(4τ)^-4 in the next section (c.f. Section <ref>) and show that it is, among others, a Hecke eigenform in a generalized sense. (In <cit.>, the authors obtained some Hecke eigenforms in the generalized sense. However, for N=4, only Hecke eigenforms of weight 1/2, 1, 3/2 are obtained. The current example gives a Hecke eigenform of weight 5/2 which can not be achieved using methods in <cit.>.)





§ APPLICATION: AN EXTENSION OF MARTIN'S LIST OF MULTIPLICATIVE ETA-QUOTIENTS


In 1996 Martin <cit.> obtained the complete list of integral-weight holomorphic eta-quotients that are eigenforms for all Hecke operators. This generalized a previous result of Dummit, Kisilevsky and McKay <cit.>. The term “eta-quotient” in <cit.> refers to one whose multiplier system is induced by a Dirichlet character (necessarily real) and the Hecke operators are in the sense of <cit.>.

In this section, we provide an algorithm for finding out all holomorphic eta-quotients f(τ)=∏_n≥1η(nτ)^r_n of integral exponents r_n (integral or half-integral weights) and of arbitrary multiplier systems such that either of the following conditions holds:

  
  * _ M_k(Γ_0(N),χ) can be computed using Theorem <ref> (with t=0) and is equal to 1,
  
  * _ M_k(Γ_0(N),χ) can be computed using Theorem <ref> (with t=0) and is greater than 1, f is a cusp form, k>2 and there exist _ M_k(Γ_0(N),χ)-1 linearly independent Eisenstein series in M_k(Γ_0(N),χ) (c.f. Definition <ref>),

where k=1/2∑_nr_n is the weight, N is the least common multiple of all n with r_n≠ 0 (when r_n=0 for all n, let N=1) and χ is the character Γ_0(N)→^× of f. (Note that throughout this section Γ_0(N)=Γ_0(N)2 as in Section <ref>.)


If f satisfies either of the above conditions, we say it is admissible. If f satisfies the condition (a) then we say it is admissible of type I; if it satisfies (b) then we say it is admissible of type II. Moreover, if f satisfies either of the conditions

  
  (a') _ M_k(Γ_0(N),χ)=1,
  
  (b') _ M_k(Γ_0(N),χ)≥2 but _ S_k(Γ_0(N),χ)=1 and f is a cusp form,

we say f is weakly admissible. Thus an admissible eta-quotient must be a weakly admissible eta-quotient. Similarly, we can define the concept of weakly admissible eta-quotients of type I and type II. For each weakly admissible f, we will show that it is a Hecke eigenform in the sense of Wohlfahrt <cit.>; c.f. Theorem <ref>. Therefore, our list of admissible eta-quotients can be considered as an extension of Martin's list.



 §.§ The algorithm for finding admissible eta-quotients


Throughout this subsection, let f(τ)=∏_n≥1η(nτ)^r_n be a holomorphic eta-quotient where r_n are integers and are equal to 0 for all but finitely many n. Let k=1/2∑_nr_n be the weight and N, the level[The reader should notice that this usage of “eta quotient of level N” is different from that in the second half of Section <ref>. In the usage here any eta-quotient has a uniquely determined level while in the usage before, an eta-quotient, say η(τ)η(2τ), may be regarded as of level 2, or of level 4, etc.], be the least common multiple of {n∈Z1 r_n≠0} (with the convention N=1 if r_n=0 for all n). Let χΓ_0(N)→^× be the character of f (c.f. (<ref>) with D=2). Since f is holomorphic at all cusps we have x_c≥0 for all c| N (x_c are defined in (<ref>)).  By Remark <ref> we have

    _ M_k(Γ_0(N), χ)   ≥k-1/12m+1/4ε_2+1/3ε_3+∑_c| Nϕ(c,N/c)·(1/2-{x_c/24})
       >k-1/12m-1/2∑_c| Nϕ(c,N/c)

where m, ε_2, ε_3 are given by (<ref>), (<ref>), (<ref>) respectively. 


If ((k-1)N/12-3√(N)/2)·∏_p| N(1+1/p)≥1, then f is not admissible. As a consequence, there are only finitely many admissible eta-quotients.


Assume by contradiction that f is admissible. If the condition (a) holds, then

    1=_ M_k(Γ_0(N), χ)>k-1/12m-1/2∑_c| Nϕ(c,N/c).

By (<ref>) we have ε_∞=∑_c| Nϕ(c,N/c)≤ m· N^-1/2. This, together with (<ref>) and (<ref>), implies that ((k-1)N/12-√(N)/2)·∏_p| N(1+1/p)<1 which contradicts the assumption. On the other hand, if the condition (b) holds. Since k>2, there are totally[These n_0 functions, in the rational weight (k>2) situation, constitute a basis of the Eisenstein space  which, as in the integral weight case, is defined as the orthogonal complement of the space of cusp forms under the Petersson inner product. In addition, one can prove that n_0=R where R is (<ref>) with f_0 arbitrarily chosen.] n_0 Eisenstein series in M_k(Γ_0(N), χ), where

    n_0=#{s ∈Γ_0(N)\χ(γ_sT^w_sγ_s^-1)=1 },

(w_s is the width; c.f. Section <ref>, γ_s is any matrix in  such that γ_s(∞)=s) and these n_0 functions are -linearly independent. See <cit.>. By condition (b) we have n_0≥_ M_k(Γ_0(N), χ)-1. Since in the current case f is a cusp form, these n_0 Eisenstein series and f together constitute a basis of M_k(Γ_0(N), χ) and n_0=_ M_k(Γ_0(N), χ)-1. Hence

    n_0+1=_ M_k(Γ_0(N), χ)>k-1/12m-1/2∑_c| Nϕ(c,N/c).

This inequality, together with (<ref>), (<ref>) and the fact n_0≤ε_∞=∑_c| Nϕ(c,N/c), implies that ((k-1)N/12-3√(N)/2)·∏_p| N(1+1/p)<1 which contradicts the assumption. Therefore, we have shown that both of conditions (a) and (b) lead to contradictions, so f is not admissible.

Now we begin to prove that there are only finitely many admissible eta-quotients. Since f∈ M_k(Γ_0(N), χ) is a holomorphic eta-quotient of integral exponents, 2k∈Z0. For k=0, we must have f=1. For k=1/2 and 1, we have shown in Remark <ref> that there are only finitely many N such that _ M_k(Γ_0(N), χ) can be calculated using Theorem <ref>. For each pair (k,N), the holomorphic eta-quotients of level N and weight k constitute a finite set since the cardinality of this set does not exceed the cardinality of the solution set of ∑_c| Nϕ(c,N/c)x_c=2mk, x_c∈Z0 (c.f. Proposition <ref> and Remark <ref>) which is finite. This proves that there are only finitely many admissible eta-quotients of weights 0, 1/2 and 1. For each weight k≥ 3/2, there exists an N_0 such that when N>N_0 we have ((k-1)N/12-3√(N)/2)·∏_p| N(1+1/p)≥1. Thus, the level of an admissible eta-quotient of weight k does not exceed N_0 and hence there are finitely many of them. Finally, if k≥19 and N≥2, we have

    ((k-1)N/12-3√(N)/2)·∏_p| N(1+1/p)≥3/2(N-√(N))·∏_p| N(1+1/p)≥1,

and if k≥31, N=1, then the left-hand side still ≥1. It follows that if the weight of f is greater than or equal to 31, then f is not admissible. Therefore, there are only finitely many admissible eta-quotients.



If N>400, then f is not admissible. Moreover, if N>36, then f is not admissible of type I.


Suppose N>400; we have k≠0. If k=1/2, 1 or 3/2, then by Tables <ref>, <ref> and Section <ref>, _ M_k(Γ_0(N), χ) can not be calculated using Theorem <ref> with t=0 (that is, the first inequality in (<ref>) does not hold) for any χ. Thus, f is not admissible[Although f is possibly weakly admissible.]. Otherwise, if k≥2, then obviously ((k-1)N/12-3√(N)/2)·∏_p| N(1+1/p)≥1. It follows from this and Lemma <ref> that f is not admissible.

Now we begin to prove the latter assertion; suppose N>36. Certainly k≠0. According to Table <ref>, Table <ref> and Subsection <ref>, if k=1/2, 1 or 3/2, then _ M_k(Γ_0(N),χ)≥2 provided it can be calculated using Theorem <ref> (with t=0). Hence f is not admissible of type I. If k≥2, then for 36<N≤400 but N≠ 49 we have

    _ M_k(Γ_0(N), χ)>k-1/12m-1/2∑_c| Nϕ(c,N/c)>1.

See Appendix <ref> for the SageMath code verifying this. If N=49, then ε_3=2 (c.f. (<ref>)), and hence

    _ M_k(Γ_0(49), χ)   ≥k-1/12m+1/4ε_2+1/3ε_3+∑_c| Nϕ(c,N/c)·(1/2-{x_c/24})
       >2-1/12·56+2/3-1/2∑_c| 49ϕ(c,49/c)>1.

Therefore f is not admissible of type I for k≥2, 36<N≤400.


According to the above corollary and Lemma <ref>, the following algorithm can be used to present all admissible nonconstant eta-quotients.

  
  * Let N range over {1,2,3,4,…,400}. 
  
  * For a fixed N, set K_N={k∈1/21/2≤ k<1+12/m+18/√(N)}, where m is defined in (<ref>). 
  
  * Let N be fixed as above. For any c| N, let x_c range over nonnegative integers such that ∑_c| Nϕ(c,N/c)x_c∈2m· K_N.
  
  * Let N and (x_c)_c| N be fixed. Determine the sequence (r_n)_n| N by (<ref>) and set k=1/2∑_n| Nr_n, g=∏_n| Nη(nτ)^r_n. Since ∑_c| Nϕ(c,N/c)x_c=m·∑_n| Nr_n (c.f. Remark <ref>) we have k∈ K_N.
  
  * If r_n are not all integers, or they are all integers but N is not the least common multiple of {r_n≠ 0 n| N}, then we continue to the next value of (x_c)_c| N in Step (d) since in these cases either g is not an eta-quotient of integral exponents or g has been considered in a previous step.
  
  * We check whether (<ref>) with t=0 holds. If it does not hold, g is not admissible so we go to Step (d) and consider the next value of (x_c)_c| N.
  
  * We check whether _ M_k(Γ_0(N), χ)=1 via (<ref>). If this is the case, then we find an admissible g. We record it, go to Step (d) and consider the next value of (x_c)_c| N.
  
  * We check whether k>2, g is a cusp form (equivalent to x_c>0 for all c| N) and _ M_k(Γ_0(N), χ)=n_0+1 where n_0 is given in (<ref>). If this is the case, then g is admissible which we record; otherwise g is not admissible. We have completely determined whether this g is admissible or not so we go to Step (d) and consider the next value of (x_c)_c| N.
  
  * Since for each N there are only finitely many values of (x_c)_c| N, this algorithm halts in finitely many steps.


The above algorithm gives all admissible nonconstant eta-quotients with no repetition.


First, it is immediate that the functions g recorded in Steps (g) or (h) are admissible nonconstant eta-quotients. Second, suppose g_1 and g_2 are functions recorded in Steps (g) or (h) corresponding to different pairs (N_1,(x_c)_c| N_1) and (N_2,(x'_c)_c| N_2). If N_1≠ N_2, then the exponents of g_1 are different from those of g_2 since their least common multiples are different. Hence g_1≠ g_2. Otherwise N_1=N_2 but their is some c| N_1 such that x_c≠ x'_c. Thus the sequences of exponents of g_1 and g_2 are different by Proposition <ref>, and hence g_1≠ g_2 as well. Finally, let f=∏_n| N_0η(nτ)^r_n be any admissible nonconstant eta-quotient whose level is N_0 and weight is k_0. By Corollary <ref> we have 1≤ N_0≤ 400; by Lemma <ref> we have k_0∈ K_N_0. Define (y_c)_c| N_0 by (<ref>) (with N=N_0 and x_c replaced by y_c). It is immediate that y_c∈. Moreover, y_c≥0 since f is holomorphic at all cusps. We have ∑_c| N_0ϕ(c,N_0/c)y_c=2mk_0 (c.f. Remark <ref>) and hence ∑_c| N_0ϕ(c,N_0/c)y_c∈2m· K_N_0 where m=N_0∏_p| N_0(1+1/p)^-1. It follows that in Steps (a) and (c) the variable N can actually take the value N_0 and (x_c)_c| N can take the value (y_c)_c| N_0. The corresponding g in Step (d) is equal to f according to Proposition <ref>. Therefore, f is recorded in some step of the algorithm which concludes the proof.


If Step (h) is removed, then the remaining part of the algorithm gives all admissible nonconstant eta-quotients of type I. If this is the task, then we can shrink the set that N ranges over in Step (a) to {1,2,3,…,36} by the latter assertion of Corollary <ref>. In the following subsections, we will only consider such eta-quotients because finding those of type II need more effort and the program takes rather long time[We have provided the SageMath code that can produce admissible eta-quotients of type II. See Appendix <ref> for the usage.] which we will do in a future paper. The simplest example of type II is η(τ)^24 since it is a cusp form, _ M_12()=2 and there is an Eisenstein series in M_12(). As a comparison, η(τ)^r is an admissible eta-quotient of type I for r=0,1,2,…,23 since _ M_r/2(,χ_η^r)=1. See <cit.>.




 §.§ The table of admissible eta-quotients of type I

For the SageMath program realizing the algorithm for finding admissible eta-quotients of type I in the last subsection, see Appendix <ref>. Here (Table <ref> below) we present all such eta-quotients for levels N=1, p_1, p_1^2, p_1^3 and p_1p_2 where p_1, p_2 are distinct primes.

llllll
All admissible eta-quotients of type I of level N=1,p_1, p_1^2, p_1^3 or p_1p_2 where p_1, p_2 are distinct primes. An entry n_1^r_1n_2^r_2⋯ n_t^r_t represents the eta-quotient η(n_1τ)^r_1η(n_2τ)^r_2…η(n_tτ)^r_t and k is the weight. 



6lTable <ref>. (continued) 



6rTo be continued 

6lN=1

6l1^r where r=0,1,2,…,23

6lN=2, k=1/2

1^-12^2     1^02^1     1^22^-1           

6lN=2, k=1

1^-22^4     1^-12^3     1^02^2     1^12^1     1^32^-1     1^42^-2

6lN=2, k=3/2

1^-32^6     1^-22^5     1^-12^4     1^02^3     1^12^2     1^22^1

1^42^-1     1^52^-2     1^62^-3           

6lN=2, k=2

1^-42^8     1^-32^7     1^-22^6     1^-12^5     1^02^4     1^12^3

1^22^2     1^32^1     1^52^-1     1^62^-2     1^72^-3     1^82^-4

6lN=2, k=5/2

1^-52^10     1^-42^9     1^-32^8     1^-22^7     1^-12^6     1^02^5

1^12^4     1^22^3     1^32^2     1^42^1     1^62^-1     1^72^-2

1^82^-3     1^92^-4     1^102^-5           

6lN=2, k=3

1^-62^12     1^-52^11     1^-42^10     1^-32^9     1^-22^8     1^-12^7

1^02^6     1^12^5     1^22^4     1^32^3     1^42^2     1^52^1

1^72^-1     1^82^-2     1^92^-3     1^102^-4     1^112^-5     1^122^-6

6lN=2, k=7/2

1^-72^14     1^-62^13     1^-52^12     1^-42^11     1^-32^10     1^-22^9

1^-12^8     1^02^7     1^12^6     1^22^5     1^32^4     1^42^3

1^52^2     1^62^1     1^82^-1     1^92^-2     1^102^-3     1^112^-4

1^122^-5     1^132^-6     1^142^-7           

6lN=2, k=4

1^-72^15     1^-62^14     1^-52^13     1^-42^12     1^-32^11     1^-22^10

1^-12^9     1^02^8     1^12^7     1^22^6     1^32^5     1^42^4

1^52^3     1^62^2     1^72^1     1^92^-1     1^102^-2     1^112^-3

1^122^-4     1^132^-5     1^142^-6     1^152^-7        

6lN=2, k=9/2

1^-52^14     1^-42^13     1^-32^12     1^-22^11     1^-12^10     1^02^9

1^12^8     1^22^7     1^32^6     1^42^5     1^52^4     1^62^3

1^72^2     1^82^1     1^102^-1     1^112^-2     1^122^-3     1^132^-4

1^142^-5                 

6lN=2, k=5

1^-32^13     1^-22^12     1^-12^11     1^02^10     1^12^9     1^22^8

1^32^7     1^42^6     1^52^5     1^62^4     1^72^3     1^82^2

1^92^1     1^112^-1     1^122^-2     1^132^-3        

6lN=2, k=11/2

1^-12^12     1^02^11     1^12^10     1^22^9     1^32^8     1^42^7

1^52^6     1^62^5     1^72^4     1^82^3     1^92^2     1^102^1

1^122^-1                 

6lN=2, k=6

1^12^11     1^22^10     1^32^9     1^42^8     1^52^7     1^62^6

1^72^5     1^82^4     1^92^3     1^102^2     1^112^1     

6lN=2, k=13/2

1^32^10     1^42^9     1^52^8     1^62^7     1^72^6     1^82^5

1^92^4     1^102^3              

6lN=2, k=7

1^52^9     1^62^8     1^72^7     1^82^6     1^92^5     

6lN=2, k=15/2

1^72^8     1^82^7              

6lN=3, k=1/2

1^03^1                 

6lN=3, k=1

1^-13^3     1^03^2     1^13^1     1^33^-1        

6lN=3, k=3/2

1^-13^4     1^03^3     1^13^2     1^23^1     1^43^-1     

6lN=3, k=2

1^-23^6     1^-13^5     1^03^4     1^13^3     1^23^2     1^33^1

1^53^-1     1^63^-2              

6lN=3, k=5/2

1^-23^7     1^-13^6     1^03^5     1^13^4     1^23^3     1^33^2

1^43^1     1^63^-1     1^73^-2           

6lN=3, k=3

1^-23^8     1^-13^7     1^03^6     1^13^5     1^23^4     1^33^3

1^43^2     1^53^1     1^73^-1     1^83^-2        

6lN=3, k=7/2

1^-13^8     1^03^7     1^13^6     1^23^5     1^33^4     1^43^3

1^53^2     1^63^1     1^83^-1           

6lN=3, k=4

1^13^7     1^23^6     1^33^5     1^43^4     1^53^3     1^63^2

1^73^1                 

6lN=3, k=9/2

1^23^7     1^33^6     1^43^5     1^53^4     1^63^3     1^73^2

6lN=3, k=5

1^43^6     1^53^5     1^63^4           

6lN=3, k=11/2

1^53^6     1^63^5              

6lN=4, k=1/2

1^-22^54^-2     1^-12^34^-1     1^02^-14^2     1^02^04^1     1^02^24^-1     1^12^-14^1

6lN=4, k=1

1^-42^104^-4     1^-32^74^-2     1^-32^84^-3     1^-22^54^-1     1^-22^64^-2     1^-22^74^-3

1^-12^14^2     1^-12^24^1     1^-12^44^-1     1^-12^54^-2     1^02^-24^4     1^02^-14^3

1^02^04^2     1^02^14^1     1^02^34^-1     1^02^44^-2     1^12^-24^3     1^12^-14^2

1^12^04^1     1^12^24^-1     1^22^-24^2     1^22^-14^1     1^22^14^-1     1^32^-24^1

6lN=4, k=3/2

1^-62^154^-6     1^-52^124^-4     1^-52^134^-5     1^-42^94^-2     1^-42^104^-3     1^-42^114^-4

1^-42^124^-5     1^-32^74^-1     1^-32^84^-2     1^-32^94^-3     1^-32^104^-4     1^-22^34^2

1^-22^44^1     1^-22^64^-1     1^-22^74^-2     1^-22^84^-3     1^-22^94^-4     1^-12^04^4

1^-12^14^3     1^-12^24^2     1^-12^34^1     1^-12^54^-1     1^-12^64^-2     1^-12^74^-3

1^02^-34^6     1^02^-24^5     1^02^-14^4     1^02^04^3     1^02^14^2     1^02^24^1

1^02^44^-1     1^02^54^-2     1^02^64^-3     1^12^-34^5     1^12^-24^4     1^12^-14^3

1^12^04^2     1^12^14^1     1^12^34^-1     1^12^44^-2     1^22^-34^4     1^22^-24^3

1^22^-14^2     1^22^04^1     1^22^24^-1     1^22^34^-2     1^32^-34^3     1^32^-24^2

1^32^-14^1     1^32^14^-1     1^42^-34^2     1^42^-24^1     1^42^04^-1     1^52^-34^1

6lN=4, k=2

1^-72^174^-6     1^-72^184^-7     1^-62^144^-4     1^-62^154^-5     1^-62^164^-6     1^-62^174^-7

1^-52^114^-2     1^-52^124^-3     1^-52^134^-4     1^-52^144^-5     1^-52^154^-6     1^-42^94^-1

1^-42^104^-2     1^-42^114^-3     1^-42^124^-4     1^-42^134^-5     1^-42^144^-6     1^-32^54^2

1^-32^64^1     1^-32^84^-1     1^-32^94^-2     1^-32^104^-3     1^-32^114^-4     1^-32^124^-5

1^-22^24^4     1^-22^34^3     1^-22^44^2     1^-22^54^1     1^-22^74^-1     1^-22^84^-2

1^-22^94^-3     1^-22^104^-4     1^-22^114^-5     1^-12^-14^6     1^-12^04^5     1^-12^14^4

1^-12^24^3     1^-12^34^2     1^-12^44^1     1^-12^64^-1     1^-12^74^-2     1^-12^84^-3

1^-12^94^-4     1^02^-34^7     1^02^-24^6     1^02^-14^5     1^02^04^4     1^02^14^3

1^02^24^2     1^02^34^1     1^02^54^-1     1^02^64^-2     1^02^74^-3     1^02^84^-4

1^12^-44^7     1^12^-34^6     1^12^-24^5     1^12^-14^4     1^12^04^3     1^12^14^2

1^12^24^1     1^12^44^-1     1^12^54^-2     1^12^64^-3     1^22^-44^6     1^22^-34^5

1^22^-24^4     1^22^-14^3     1^22^04^2     1^22^14^1     1^22^34^-1     1^22^44^-2

1^22^54^-3     1^32^-44^5     1^32^-34^4     1^32^-24^3     1^32^-14^2     1^32^04^1

1^32^24^-1     1^32^34^-2     1^42^-44^4     1^42^-34^3     1^42^-24^2     1^42^-14^1

1^42^14^-1     1^42^24^-2     1^52^-44^3     1^52^-34^2     1^52^-24^1     1^52^04^-1

1^62^-44^2     1^62^-34^1     1^62^-14^-1     1^72^-44^1        

6lN=4, k=5/2

1^-72^164^-4     1^-72^174^-5     1^-72^184^-6     1^-62^134^-2     1^-62^144^-3     1^-62^154^-4

1^-62^164^-5     1^-62^174^-6     1^-62^184^-7     1^-52^114^-1     1^-52^124^-2     1^-52^134^-3

1^-52^144^-4     1^-52^154^-5     1^-52^164^-6     1^-52^174^-7     1^-42^74^2     1^-42^84^1

1^-42^104^-1     1^-42^114^-2     1^-42^124^-3     1^-42^134^-4     1^-42^144^-5     1^-42^154^-6

1^-42^164^-7     1^-32^44^4     1^-32^54^3     1^-32^64^2     1^-32^74^1     1^-32^94^-1

1^-32^104^-2     1^-32^114^-3     1^-32^124^-4     1^-32^134^-5     1^-32^144^-6     1^-22^24^5

1^-22^34^4     1^-22^44^3     1^-22^54^2     1^-22^64^1     1^-22^84^-1     1^-22^94^-2

1^-22^104^-3     1^-22^114^-4     1^-22^124^-5     1^-22^134^-6     1^-12^04^6     1^-12^14^5

1^-12^24^4     1^-12^34^3     1^-12^44^2     1^-12^54^1     1^-12^74^-1     1^-12^84^-2

1^-12^94^-3     1^-12^104^-4     1^-12^114^-5     1^02^-14^6     1^02^04^5     1^02^14^4

1^02^24^3     1^02^34^2     1^02^44^1     1^02^64^-1     1^02^74^-2     1^02^84^-3

1^02^94^-4     1^02^104^-5     1^12^-34^7     1^12^-24^6     1^12^-14^5     1^12^04^4

1^12^14^3     1^12^24^2     1^12^34^1     1^12^54^-1     1^12^64^-2     1^12^74^-3

1^12^84^-4     1^22^-44^7     1^22^-34^6     1^22^-24^5     1^22^-14^4     1^22^04^3

1^22^14^2     1^22^24^1     1^22^44^-1     1^22^54^-2     1^22^64^-3     1^22^74^-4

1^32^-54^7     1^32^-44^6     1^32^-34^5     1^32^-24^4     1^32^-14^3     1^32^04^2

1^32^14^1     1^32^34^-1     1^32^44^-2     1^32^54^-3     1^42^-54^6     1^42^-44^5

1^42^-34^4     1^42^-24^3     1^42^-14^2     1^42^04^1     1^42^24^-1     1^42^34^-2

1^42^44^-3     1^52^-54^5     1^52^-44^4     1^52^-34^3     1^52^-24^2     1^52^-14^1

1^52^14^-1     1^52^24^-2     1^62^-54^4     1^62^-44^3     1^62^-34^2     1^62^-24^1

1^62^04^-1     1^72^-54^3     1^72^-44^2     1^72^-34^1        

6lN=4, k=3

1^-72^154^-2     1^-72^164^-3     1^-72^174^-4     1^-62^134^-1     1^-62^144^-2     1^-62^154^-3

1^-62^164^-4     1^-62^174^-5     1^-52^94^2     1^-52^104^1     1^-52^124^-1     1^-52^134^-2

1^-52^144^-3     1^-52^154^-4     1^-52^164^-5     1^-52^174^-6     1^-42^74^3     1^-42^84^2

1^-42^94^1     1^-42^114^-1     1^-42^124^-2     1^-42^134^-3     1^-42^144^-4     1^-42^154^-5

1^-42^164^-6     1^-42^174^-7     1^-32^54^4     1^-32^64^3     1^-32^74^2     1^-32^84^1

1^-32^104^-1     1^-32^114^-2     1^-32^124^-3     1^-32^134^-4     1^-32^144^-5     1^-32^154^-6

1^-32^164^-7     1^-22^44^4     1^-22^54^3     1^-22^64^2     1^-22^74^1     1^-22^94^-1

1^-22^104^-2     1^-22^114^-3     1^-22^124^-4     1^-22^134^-5     1^-22^144^-6     1^-22^154^-7

1^-12^24^5     1^-12^34^4     1^-12^44^3     1^-12^54^2     1^-12^64^1     1^-12^84^-1

1^-12^94^-2     1^-12^104^-3     1^-12^114^-4     1^-12^124^-5     1^-12^134^-6     1^02^14^5

1^02^24^4     1^02^34^3     1^02^44^2     1^02^54^1     1^02^74^-1     1^02^84^-2

1^02^94^-3     1^02^104^-4     1^02^114^-5     1^02^124^-6     1^12^-14^6     1^12^04^5

1^12^14^4     1^12^24^3     1^12^34^2     1^12^44^1     1^12^64^-1     1^12^74^-2

1^12^84^-3     1^12^94^-4     1^12^104^-5     1^22^-24^6     1^22^-14^5     1^22^04^4

1^22^14^3     1^22^24^2     1^22^34^1     1^22^54^-1     1^22^64^-2     1^22^74^-3

1^22^84^-4     1^22^94^-5     1^32^-44^7     1^32^-34^6     1^32^-24^5     1^32^-14^4

1^32^04^3     1^32^14^2     1^32^24^1     1^32^44^-1     1^32^54^-2     1^32^64^-3

1^32^74^-4     1^42^-54^7     1^42^-44^6     1^42^-34^5     1^42^-24^4     1^42^-14^3

1^42^04^2     1^42^14^1     1^42^34^-1     1^42^44^-2     1^42^54^-3     1^52^-64^7

1^52^-54^6     1^52^-44^5     1^52^-34^4     1^52^-24^3     1^52^-14^2     1^52^04^1

1^52^24^-1     1^62^-64^6     1^62^-54^5     1^62^-44^4     1^62^-34^3     1^62^-24^2

1^62^-14^1     1^72^-64^5     1^72^-54^4     1^72^-44^3        

6lN=4, k=7/2

1^-72^154^-1     1^-72^164^-2     1^-62^124^1     1^-62^144^-1     1^-62^154^-2     1^-62^164^-3

1^-52^104^2     1^-52^114^1     1^-52^134^-1     1^-52^144^-2     1^-52^154^-3     1^-52^164^-4

1^-42^94^2     1^-42^104^1     1^-42^124^-1     1^-42^134^-2     1^-42^144^-3     1^-42^154^-4

1^-42^164^-5     1^-32^74^3     1^-32^84^2     1^-32^94^1     1^-32^114^-1     1^-32^124^-2

1^-32^134^-3     1^-32^144^-4     1^-32^154^-5     1^-32^164^-6     1^-22^64^3     1^-22^74^2

1^-22^84^1     1^-22^104^-1     1^-22^114^-2     1^-22^124^-3     1^-22^134^-4     1^-22^144^-5

1^-22^154^-6     1^-22^164^-7     1^-12^44^4     1^-12^54^3     1^-12^64^2     1^-12^74^1

1^-12^94^-1     1^-12^104^-2     1^-12^114^-3     1^-12^124^-4     1^-12^134^-5     1^-12^144^-6

1^-12^154^-7     1^02^34^4     1^02^44^3     1^02^54^2     1^02^64^1     1^02^84^-1

1^02^94^-2     1^02^104^-3     1^02^114^-4     1^02^124^-5     1^02^134^-6     1^02^144^-7

1^12^14^5     1^12^24^4     1^12^34^3     1^12^44^2     1^12^54^1     1^12^74^-1

1^12^84^-2     1^12^94^-3     1^12^104^-4     1^12^114^-5     1^12^124^-6     1^22^04^5

1^22^14^4     1^22^24^3     1^22^34^2     1^22^44^1     1^22^64^-1     1^22^74^-2

1^22^84^-3     1^22^94^-4     1^22^104^-5     1^32^-24^6     1^32^-14^5     1^32^04^4

1^32^14^3     1^32^24^2     1^32^34^1     1^32^54^-1     1^32^64^-2     1^32^74^-3

1^42^-34^6     1^42^-24^5     1^42^-14^4     1^42^04^3     1^42^14^2     1^42^24^1

1^42^44^-1     1^52^-54^7     1^52^-44^6     1^52^-34^5     1^52^-24^4     1^52^-14^3

1^52^04^2     1^52^14^1     1^62^-64^7     1^62^-54^6     1^62^-44^5     1^62^-34^4

1^62^-24^3     1^72^-74^7     1^72^-64^6     1^72^-54^5        

6lN=4, k=4

1^-62^154^-1     1^-52^124^1     1^-52^144^-1     1^-52^154^-2     1^-42^114^1     1^-42^134^-1

1^-42^144^-2     1^-42^154^-3     1^-32^94^2     1^-32^104^1     1^-32^124^-1     1^-32^134^-2

1^-32^144^-3     1^-32^154^-4     1^-22^84^2     1^-22^94^1     1^-22^114^-1     1^-22^124^-2

1^-22^134^-3     1^-22^144^-4     1^-22^154^-5     1^-12^64^3     1^-12^74^2     1^-12^84^1

1^-12^104^-1     1^-12^114^-2     1^-12^124^-3     1^-12^134^-4     1^-12^144^-5     1^-12^154^-6

1^02^54^3     1^02^64^2     1^02^74^1     1^02^94^-1     1^02^104^-2     1^02^114^-3

1^02^124^-4     1^02^134^-5     1^02^144^-6     1^02^154^-7     1^12^34^4     1^12^44^3

1^12^54^2     1^12^64^1     1^12^84^-1     1^12^94^-2     1^12^104^-3     1^12^114^-4

1^12^124^-5     1^22^24^4     1^22^34^3     1^22^44^2     1^22^54^1     1^22^74^-1

1^22^84^-2     1^22^94^-3     1^32^04^5     1^32^14^4     1^32^24^3     1^32^34^2

1^32^44^1     1^32^64^-1     1^42^-14^5     1^42^04^4     1^42^14^3     1^42^24^2

1^42^34^1     1^52^-34^6     1^52^-24^5     1^52^-14^4     1^52^04^3     1^62^-44^6

1^62^-34^5     1^72^-64^7              

6lN=4, k=9/2

1^-42^144^-1     1^-32^114^1     1^-32^134^-1     1^-32^144^-2     1^-22^104^1     1^-22^124^-1

1^-22^134^-2     1^-22^144^-3     1^-12^84^2     1^-12^94^1     1^-12^114^-1     1^-12^124^-2

1^-12^134^-3     1^-12^144^-4     1^02^74^2     1^02^84^1     1^02^104^-1     1^02^114^-2

1^02^124^-3     1^02^134^-4     1^02^144^-5     1^12^54^3     1^12^64^2     1^12^74^1

1^12^94^-1     1^12^104^-2     1^12^114^-3     1^22^44^3     1^22^54^2     1^22^64^1

1^22^84^-1     1^32^24^4     1^32^34^3     1^32^44^2     1^32^54^1     1^42^14^4

1^42^24^3     1^52^-14^5              

6lN=4, k=5

1^-22^134^-1     1^-12^104^1     1^-12^124^-1     1^-12^134^-2     1^02^94^1     1^02^114^-1

1^02^124^-2     1^02^134^-3     1^12^74^2     1^12^84^1     1^12^104^-1     1^22^64^2

1^22^74^1     1^32^44^3              

6lN=4, k=11/2

1^02^124^-1     1^12^94^1              

6lN=5, k=1/2

1^05^1                 

6lN=5, k=1

1^05^2     1^15^1              

6lN=5, k=3/2

1^05^3     1^15^2     1^25^1           

6lN=5, k=2

1^05^4     1^15^3     1^25^2     1^35^1        

6lN=5, k=5/2

1^15^4     1^25^3     1^35^2     1^45^1        

6lN=5, k=3

1^25^4     1^35^3     1^45^2           

6lN=5, k=7/2

1^35^4     1^45^3              

6lN=6, k=1/2

1^-12^13^26^-1     1^-12^23^16^-1     1^02^03^-16^2     1^02^03^06^1     1^02^03^26^-1     1^12^-13^-16^2

1^22^-13^-16^1                 

6lN=6, k=1

1^-22^23^46^-2     1^-22^33^26^-1     1^-22^33^36^-2     1^-22^43^16^-1     1^-22^43^26^-2     1^-12^13^16^1

1^-12^13^26^0     1^-12^13^36^-1     1^-12^13^46^-2     1^-12^23^-16^2     1^-12^23^06^1     1^-12^23^16^0

1^-12^23^26^-1     1^-12^23^36^-2     1^-12^33^16^-1     1^02^-13^06^3     1^02^03^-26^4     1^02^03^-16^3

1^02^03^06^2     1^02^03^16^1     1^02^03^36^-1     1^02^03^46^-2     1^02^13^-16^2     1^02^13^06^1

1^02^13^16^0     1^02^13^26^-1     1^02^23^16^-1     1^02^33^06^-1     1^12^-13^-26^4     1^12^-13^-16^3

1^12^-13^06^2     1^12^-13^16^1     1^12^03^-16^2     1^12^03^06^1     1^12^03^26^-1     1^12^13^-16^1

1^12^13^16^-1     1^22^-23^-26^4     1^22^-13^-26^3     1^22^-13^-16^2     1^22^-13^06^1     1^22^-13^16^0

1^22^-13^26^-1     1^22^03^-16^1     1^32^-23^-26^3     1^32^-23^-16^2     1^32^-13^-16^1     1^42^-23^-26^2

1^42^-23^-16^1                 

6lN=6, k=3/2

1^-32^43^46^-2     1^-32^43^56^-3     1^-32^53^26^-1     1^-32^53^36^-2     1^-32^53^46^-3     1^-22^23^36^0

1^-22^23^46^-1     1^-22^23^56^-2     1^-22^33^16^1     1^-22^33^26^0     1^-22^33^36^-1     1^-22^33^46^-2

1^-22^33^56^-3     1^-22^43^-16^2     1^-22^43^06^1     1^-22^43^16^0     1^-22^43^26^-1     1^-22^43^36^-2

1^-22^43^46^-3     1^-22^53^16^-1     1^-22^53^26^-2     1^-12^03^26^2     1^-12^03^36^1     1^-12^13^06^3

1^-12^13^16^2     1^-12^13^26^1     1^-12^13^36^0     1^-12^13^46^-1     1^-12^13^56^-2     1^-12^23^-26^4

1^-12^23^-16^3     1^-12^23^06^2     1^-12^23^16^1     1^-12^23^26^0     1^-12^23^36^-1     1^-12^23^46^-2

1^-12^23^56^-3     1^-12^33^-16^2     1^-12^33^06^1     1^-12^33^16^0     1^-12^33^26^-1     1^-12^33^36^-2

1^-12^43^16^-1     1^-12^43^26^-2     1^02^-13^06^4     1^02^-13^16^3     1^02^-13^26^2     1^02^03^-16^4

1^02^03^06^3     1^02^03^16^2     1^02^03^26^1     1^02^03^46^-1     1^02^13^-26^4     1^02^13^-16^3

1^02^13^06^2     1^02^13^16^1     1^02^13^26^0     1^02^13^36^-1     1^02^13^46^-2     1^02^23^-16^2

1^02^23^06^1     1^02^23^16^0     1^02^23^26^-1     1^02^23^36^-2     1^02^33^-16^1     1^02^33^16^-1

1^02^33^26^-2     1^02^43^06^-1     1^12^-13^-26^5     1^12^-13^-16^4     1^12^-13^06^3     1^12^-13^16^2

1^12^-13^26^1     1^12^-13^36^0     1^12^03^-26^4     1^12^03^-16^3     1^12^03^06^2     1^12^03^16^1

1^12^03^36^-1     1^12^03^46^-2     1^12^13^-26^3     1^12^13^-16^2     1^12^13^06^1     1^12^13^16^0

1^12^13^26^-1     1^12^13^36^-2     1^12^23^-16^1     1^12^23^16^-1     1^12^33^06^-1     1^22^-23^-26^5

1^22^-23^-16^4     1^22^-23^06^3     1^22^-13^-36^5     1^22^-13^-26^4     1^22^-13^-16^3     1^22^-13^06^2

1^22^-13^16^1     1^22^-13^26^0     1^22^-13^36^-1     1^22^-13^46^-2     1^22^03^-26^3     1^22^03^-16^2

1^22^03^06^1     1^22^03^26^-1     1^22^13^-16^1     1^22^13^16^-1     1^22^23^-16^0     1^22^23^06^-1

1^32^-23^-36^5     1^32^-23^-26^4     1^32^-23^-16^3     1^32^-23^06^2     1^32^-23^16^1     1^32^-13^-26^3

1^32^-13^-16^2     1^32^-13^06^1     1^32^-13^16^0     1^32^-13^26^-1     1^32^03^-26^2     1^32^03^-16^1

1^32^03^16^-1     1^32^13^-16^0     1^42^-33^-36^5     1^42^-33^-26^4     1^42^-23^-36^4     1^42^-23^-26^3

1^42^-23^-16^2     1^42^-23^06^1     1^42^-23^16^0     1^42^-23^26^-1     1^42^-13^-26^2     1^42^-13^-16^1

1^52^-33^-36^4     1^52^-33^-26^3     1^52^-33^-16^2     1^52^-23^-26^2     1^52^-23^-16^1     

6lN=6, k=2

1^-42^63^56^-3     1^-42^63^66^-4     1^-32^43^36^0     1^-32^43^46^-1     1^-32^43^56^-2     1^-32^53^26^0

1^-32^53^36^-1     1^-32^53^46^-2     1^-32^53^56^-3     1^-32^53^66^-4     1^-22^23^26^2     1^-22^23^36^1

1^-22^23^46^0     1^-22^33^06^3     1^-22^33^16^2     1^-22^33^26^1     1^-22^33^36^0     1^-22^33^46^-1

1^-22^33^56^-2     1^-22^43^-16^3     1^-22^43^06^2     1^-22^43^16^1     1^-22^43^26^0     1^-22^43^36^-1

1^-22^43^46^-2     1^-22^43^56^-3     1^-22^53^36^-2     1^-22^53^46^-3     1^-12^03^26^3     1^-12^03^36^2

1^-12^13^16^3     1^-12^13^26^2     1^-12^13^36^1     1^-12^13^46^0     1^-12^23^06^3     1^-12^23^16^2

1^-12^23^26^1     1^-12^23^36^0     1^-12^23^46^-1     1^-12^33^-26^4     1^-12^33^-16^3     1^-12^33^06^2

1^-12^33^16^1     1^-12^33^26^0     1^-12^33^36^-1     1^-12^33^46^-2     1^-12^33^56^-3     1^-12^43^06^1

1^-12^43^16^0     1^-12^43^26^-1     1^-12^43^36^-2     1^-12^43^46^-3     1^02^-13^26^3     1^02^-13^36^2

1^02^03^16^3     1^02^03^26^2     1^02^03^36^1     1^02^13^-16^4     1^02^13^06^3     1^02^13^16^2

1^02^13^26^1     1^02^13^36^0     1^02^13^46^-1     1^02^23^-26^4     1^02^23^-16^3     1^02^23^06^2

1^02^23^16^1     1^02^23^26^0     1^02^23^36^-1     1^02^23^46^-2     1^02^23^56^-3     1^02^33^-26^3

1^02^33^-16^2     1^02^33^06^1     1^02^33^16^0     1^02^33^26^-1     1^02^33^36^-2     1^02^33^46^-3

1^02^43^16^-1     1^02^43^26^-2     1^12^-13^06^4     1^12^-13^16^3     1^12^-13^26^2     1^12^-13^36^1

1^12^03^-16^4     1^12^03^06^3     1^12^03^16^2     1^12^03^26^1     1^12^03^46^-1     1^12^13^-26^4

1^12^13^-16^3     1^12^13^06^2     1^12^13^16^1     1^12^13^26^0     1^12^13^36^-1     1^12^13^46^-2

1^12^23^-26^3     1^12^23^-16^2     1^12^23^06^1     1^12^23^16^0     1^12^23^26^-1     1^12^23^36^-2

1^12^33^-16^1     1^12^33^16^-1     1^12^33^26^-2     1^22^-23^06^4     1^22^-23^16^3     1^22^-23^26^2

1^22^-13^-16^4     1^22^-13^06^3     1^22^-13^16^2     1^22^-13^26^1     1^22^-13^36^0     1^22^03^-36^5

1^22^03^-26^4     1^22^03^-16^3     1^22^03^06^2     1^22^03^16^1     1^22^03^36^-1     1^22^03^46^-2

1^22^13^-26^3     1^22^13^-16^2     1^22^13^06^1     1^22^13^16^0     1^22^13^26^-1     1^22^13^36^-2

1^22^23^-26^2     1^22^23^-16^1     1^22^23^16^-1     1^22^23^26^-2     1^22^33^-16^0     1^22^33^06^-1

1^32^-23^-26^5     1^32^-23^-16^4     1^32^-23^06^3     1^32^-23^16^2     1^32^-23^26^1     1^32^-23^36^0

1^32^-13^-36^5     1^32^-13^-26^4     1^32^-13^-16^3     1^32^-13^06^2     1^32^-13^16^1     1^32^-13^26^0

1^32^-13^36^-1     1^32^-13^46^-2     1^32^03^-36^4     1^32^03^-26^3     1^32^03^-16^2     1^32^03^06^1

1^32^03^26^-1     1^32^03^36^-2     1^32^13^-26^2     1^32^13^-16^1     1^32^13^16^-1     1^32^23^-16^0

1^32^23^06^-1     1^42^-33^-26^5     1^42^-33^-16^4     1^42^-33^06^3     1^42^-23^-36^5     1^42^-23^-26^4

1^42^-23^-16^3     1^42^-23^06^2     1^42^-23^16^1     1^42^-23^26^0     1^42^-23^36^-1     1^42^-13^-36^4

1^42^-13^-26^3     1^42^-13^-16^2     1^42^-13^06^1     1^42^-13^16^0     1^42^03^-26^2     1^42^03^-16^1

1^52^-33^-46^6     1^52^-33^-36^5     1^52^-33^-26^4     1^52^-33^-16^3     1^52^-33^06^2     1^52^-23^-36^4

1^52^-23^-26^3     1^62^-43^-46^6     1^62^-43^-36^5           

6lN=6, k=5/2

1^-32^43^26^2     1^-32^43^36^1     1^-32^43^46^0     1^-32^53^46^-1     1^-32^53^56^-2     1^-22^23^36^2

1^-22^33^26^2     1^-22^33^36^1     1^-22^33^46^0     1^-22^43^16^2     1^-22^43^26^1     1^-22^43^36^0

1^-22^43^46^-1     1^-22^53^56^-3     1^-12^13^36^2     1^-12^23^26^2     1^-12^23^36^1     1^-12^33^06^3

1^-12^33^16^2     1^-12^33^26^1     1^-12^33^36^0     1^-12^33^46^-1     1^-12^43^26^0     1^-12^43^36^-1

1^-12^43^46^-2     1^-12^43^56^-3     1^02^13^16^3     1^02^13^26^2     1^02^13^36^1     1^02^23^06^3

1^02^23^16^2     1^02^23^26^1     1^02^23^36^0     1^02^23^46^-1     1^02^33^-16^3     1^02^33^06^2

1^02^33^16^1     1^02^33^26^0     1^02^33^36^-1     1^02^33^46^-2     1^02^43^36^-2     1^02^43^46^-3

1^12^-13^26^3     1^12^03^16^3     1^12^03^26^2     1^12^03^36^1     1^12^13^06^3     1^12^13^16^2

1^12^13^26^1     1^12^13^36^0     1^12^23^-26^4     1^12^23^-16^3     1^12^23^06^2     1^12^23^16^1

1^12^23^26^0     1^12^23^36^-1     1^12^23^46^-2     1^12^33^06^1     1^12^33^16^0     1^12^33^26^-1

1^12^33^36^-2     1^12^33^46^-3     1^22^-23^26^3     1^22^-13^16^3     1^22^-13^26^2     1^22^03^-16^4

1^22^03^06^3     1^22^03^16^2     1^22^03^26^1     1^22^13^-26^4     1^22^13^-16^3     1^22^13^06^2

1^22^13^16^1     1^22^13^26^0     1^22^13^36^-1     1^22^13^46^-2     1^22^23^-36^4     1^22^23^-26^3

1^22^23^-16^2     1^22^23^06^1     1^22^23^16^0     1^22^23^26^-1     1^22^23^36^-2     1^22^23^46^-3

1^22^33^16^-1     1^22^33^26^-2     1^32^-23^06^4     1^32^-23^16^3     1^32^-23^26^2     1^32^-13^-16^4

1^32^-13^06^3     1^32^-13^16^2     1^32^-13^26^1     1^32^-13^36^0     1^32^03^-26^4     1^32^03^-16^3

1^32^03^06^2     1^32^03^16^1     1^32^03^36^-1     1^32^13^-36^4     1^32^13^-26^3     1^32^13^-16^2

1^32^13^06^1     1^32^13^16^0     1^32^23^-26^2     1^32^23^-16^1     1^42^-33^06^4     1^42^-33^16^3

1^42^-33^26^2     1^42^-23^-16^4     1^42^-23^06^3     1^42^-23^16^2     1^42^-23^26^1     1^42^-13^-36^5

1^42^-13^-26^4     1^42^-13^-16^3     1^42^-13^06^2     1^42^03^-36^4     1^42^03^-26^3     1^52^-33^-26^5

1^52^-33^-16^4     1^52^-23^-36^5              

6lN=6, k=3

1^-22^43^36^1     1^-12^33^26^2     1^-12^33^36^1     1^-12^43^46^-1     1^02^23^26^2     1^02^23^36^1

1^02^33^16^2     1^02^33^26^1     1^02^33^36^0     1^12^13^26^2     1^12^23^06^3     1^12^23^16^2

1^12^23^26^1     1^12^23^36^0     1^12^33^26^0     1^12^33^36^-1     1^12^33^46^-2     1^22^03^16^3

1^22^03^26^2     1^22^13^06^3     1^22^13^16^2     1^22^13^26^1     1^22^13^36^0     1^22^23^-16^3

1^22^23^06^2     1^22^23^16^1     1^22^23^26^0     1^22^23^36^-1     1^32^-13^16^3     1^32^-13^26^2

1^32^03^06^3     1^32^03^16^2     1^32^03^26^1     1^32^13^-26^4     1^32^13^-16^3     1^32^13^06^2

1^42^-23^16^3     1^42^-13^-16^4              

6lN=6, k=7/2

1^12^23^26^2     1^22^13^26^2     1^22^23^16^2     1^22^23^26^1        

6lN=7, k=1/2

1^07^1                 

6lN=7, k=1

1^07^2     1^17^1              

6lN=7, k=3/2

1^07^3     1^17^2     1^27^1           

6lN=7, k=2

1^17^3     1^27^2     1^37^1           

6lN=7, k=5/2

1^27^3     1^37^2              

6lN=8, k=1/2

1^02^-24^58^-2     1^02^-14^38^-1     1^02^04^-18^2     1^02^04^08^1     1^02^04^28^-1     1^02^14^-18^1

6lN=8, k=1

1^-22^34^38^-2     1^-22^44^18^-1     1^-22^54^-38^2     1^-22^54^-28^1     1^-22^54^08^-1     1^-22^64^-38^1

1^-12^04^58^-2     1^-12^14^38^-1     1^-12^14^48^-2     1^-12^24^-18^2     1^-12^24^08^1     1^-12^24^28^-1

1^-12^34^-28^2     1^-12^34^-18^1     1^-12^34^18^-1     1^-12^44^-28^1     1^02^-34^78^-2     1^02^-34^88^-3

1^02^-24^58^-1     1^02^-24^68^-2     1^02^-24^78^-3     1^02^-14^18^2     1^02^-14^28^1     1^02^-14^48^-1

1^02^-14^58^-2     1^02^04^-18^3     1^02^04^08^2     1^02^04^18^1     1^02^04^38^-1     1^02^04^48^-2

1^02^14^-28^3     1^02^14^-18^2     1^02^14^08^1     1^02^14^28^-1     1^02^24^-28^2     1^02^24^-18^1

1^02^24^18^-1     1^02^34^-28^1     1^12^-34^68^-2     1^12^-24^48^-1     1^12^-24^58^-2     1^12^-14^08^2

1^12^-14^18^1     1^12^-14^38^-1     1^12^04^-18^2     1^12^04^08^1     1^12^04^28^-1     1^12^14^-18^1

1^22^-34^58^-2     1^22^-24^38^-1     1^22^-14^-18^2     1^22^-14^08^1     1^22^-14^28^-1     1^22^04^-18^1

6lN=8, k=3/2

1^-32^54^38^-2     1^-32^64^18^-1     1^-32^64^28^-2     1^-32^74^-38^2     1^-32^74^-28^1     1^-32^74^08^-1

1^-32^84^-48^2     1^-32^84^-38^1     1^-32^84^-18^-1     1^-32^94^-48^1     1^-22^24^58^-2     1^-22^24^68^-3

1^-22^34^38^-1     1^-22^34^48^-2     1^-22^34^58^-3     1^-22^44^-18^2     1^-22^44^08^1     1^-22^44^28^-1

1^-22^44^38^-2     1^-22^54^-38^3     1^-22^54^-28^2     1^-22^54^-18^1     1^-22^54^18^-1     1^-22^54^28^-2

1^-22^64^-48^3     1^-22^64^-38^2     1^-22^64^-28^1     1^-22^64^08^-1     1^-22^74^-48^2     1^-22^74^-38^1

1^-22^74^-18^-1     1^-22^84^-48^1     1^-12^-14^78^-2     1^-12^-14^88^-3     1^-12^04^58^-1     1^-12^04^68^-2

1^-12^04^78^-3     1^-12^14^18^2     1^-12^14^28^1     1^-12^14^48^-1     1^-12^14^58^-2     1^-12^14^68^-3

1^-12^24^-18^3     1^-12^24^08^2     1^-12^24^18^1     1^-12^24^38^-1     1^-12^24^48^-2     1^-12^34^-28^3

1^-12^34^-18^2     1^-12^34^08^1     1^-12^34^28^-1     1^-12^34^38^-2     1^-12^44^-38^3     1^-12^44^-28^2

1^-12^44^-18^1     1^-12^44^18^-1     1^-12^54^-38^2     1^-12^54^-28^1     1^-12^54^08^-1     1^-12^64^-38^1

1^02^-34^78^-1     1^02^-34^88^-2     1^02^-24^48^1     1^02^-24^68^-1     1^02^-24^78^-2     1^02^-24^88^-3

1^02^-14^28^2     1^02^-14^38^1     1^02^-14^58^-1     1^02^-14^68^-2     1^02^-14^78^-3     1^02^04^18^2

1^02^04^28^1     1^02^04^48^-1     1^02^04^58^-2     1^02^04^68^-3     1^02^14^-18^3     1^02^14^08^2

1^02^14^18^1     1^02^14^38^-1     1^02^14^48^-2     1^02^24^-28^3     1^02^24^-18^2     1^02^24^08^1

1^02^24^28^-1     1^02^24^38^-2     1^02^34^-38^3     1^02^34^-28^2     1^02^34^-18^1     1^02^34^18^-1

1^02^44^-38^2     1^02^44^-28^1     1^02^44^08^-1     1^02^54^-38^1     1^12^-44^88^-2     1^12^-44^98^-3

1^12^-34^68^-1     1^12^-34^78^-2     1^12^-34^88^-3     1^12^-24^28^2     1^12^-24^38^1     1^12^-24^58^-1

1^12^-24^68^-2     1^12^-24^78^-3     1^12^-14^08^3     1^12^-14^18^2     1^12^-14^28^1     1^12^-14^48^-1

1^12^-14^58^-2     1^12^04^-18^3     1^12^04^08^2     1^12^04^18^1     1^12^04^38^-1     1^12^04^48^-2

1^12^14^-28^3     1^12^14^-18^2     1^12^14^08^1     1^12^14^28^-1     1^12^24^-28^2     1^12^24^-18^1

1^12^24^18^-1     1^12^34^-28^1     1^22^-44^78^-2     1^22^-44^88^-3     1^22^-34^58^-1     1^22^-34^68^-2

1^22^-34^78^-3     1^22^-24^18^2     1^22^-24^28^1     1^22^-24^48^-1     1^22^-24^58^-2     1^22^-14^-18^3

1^22^-14^08^2     1^22^-14^18^1     1^22^-14^38^-1     1^22^-14^48^-2     1^22^04^-28^3     1^22^04^-18^2

1^22^04^08^1     1^22^04^28^-1     1^22^14^-28^2     1^22^14^-18^1     1^22^14^18^-1     1^22^24^-28^1

1^32^-44^68^-2     1^32^-34^48^-1     1^32^-34^58^-2     1^32^-24^08^2     1^32^-24^18^1     1^32^-24^38^-1

1^32^-14^-18^2     1^32^-14^08^1     1^32^-14^28^-1     1^32^04^-18^1        

6lN=8, k=2

1^-32^44^58^-2     1^-32^44^68^-3     1^-32^54^38^-1     1^-32^54^48^-2     1^-32^54^58^-3     1^-32^64^-18^2

1^-32^64^08^1     1^-32^64^28^-1     1^-32^64^38^-2     1^-32^64^48^-3     1^-32^74^-38^3     1^-32^74^-28^2

1^-32^74^-18^1     1^-32^74^18^-1     1^-32^84^-48^3     1^-32^84^-38^2     1^-32^84^-28^1     1^-32^94^-58^3

1^-22^24^58^-1     1^-22^24^68^-2     1^-22^34^28^1     1^-22^34^48^-1     1^-22^34^58^-2     1^-22^34^68^-3

1^-22^44^08^2     1^-22^44^18^1     1^-22^44^38^-1     1^-22^44^48^-2     1^-22^44^58^-3     1^-22^54^-18^2

1^-22^54^08^1     1^-22^54^28^-1     1^-22^54^38^-2     1^-22^54^48^-3     1^-22^64^-38^3     1^-22^64^-28^2

1^-22^64^-18^1     1^-22^64^18^-1     1^-22^64^28^-2     1^-22^74^-48^3     1^-22^74^-38^2     1^-22^74^-28^1

1^-22^74^08^-1     1^-22^84^-58^3     1^-22^84^-48^2     1^-22^84^-38^1     1^-12^-14^78^-1     1^-12^04^48^1

1^-12^04^68^-1     1^-12^04^78^-2     1^-12^14^38^1     1^-12^14^58^-1     1^-12^14^68^-2     1^-12^14^78^-3

1^-12^24^18^2     1^-12^24^28^1     1^-12^24^48^-1     1^-12^24^58^-2     1^-12^24^68^-3     1^-12^34^08^2

1^-12^34^18^1     1^-12^34^38^-1     1^-12^34^48^-2     1^-12^34^58^-3     1^-12^44^-28^3     1^-12^44^-18^2

1^-12^44^08^1     1^-12^44^28^-1     1^-12^44^38^-2     1^-12^54^-38^3     1^-12^54^-28^2     1^-12^54^-18^1

1^-12^54^18^-1     1^-12^54^28^-2     1^-12^64^-48^3     1^-12^64^-38^2     1^-12^64^-28^1     1^-12^64^08^-1

1^-12^74^-48^2     1^-12^74^-38^1     1^-12^74^-18^-1     1^-12^84^-48^1     1^02^-24^78^-1     1^02^-14^48^1

1^02^-14^68^-1     1^02^-14^78^-2     1^02^04^38^1     1^02^04^58^-1     1^02^04^68^-2     1^02^04^78^-3

1^02^14^18^2     1^02^14^28^1     1^02^14^48^-1     1^02^14^58^-2     1^02^14^68^-3     1^02^24^08^2

1^02^24^18^1     1^02^24^38^-1     1^02^24^48^-2     1^02^24^58^-3     1^02^34^-28^3     1^02^34^-18^2

1^02^34^08^1     1^02^34^28^-1     1^02^34^38^-2     1^02^44^-38^3     1^02^44^-28^2     1^02^44^-18^1

1^02^44^18^-1     1^02^44^28^-2     1^02^54^-48^3     1^02^54^-38^2     1^02^54^-28^1     1^02^54^08^-1

1^02^64^-48^2     1^02^64^-38^1     1^02^64^-18^-1     1^02^74^-48^1     1^12^-44^88^-1     1^12^-34^58^1

1^12^-34^78^-1     1^12^-34^88^-2     1^12^-24^48^1     1^12^-24^68^-1     1^12^-24^78^-2     1^12^-24^88^-3

1^12^-14^28^2     1^12^-14^38^1     1^12^-14^58^-1     1^12^-14^68^-2     1^12^-14^78^-3     1^12^04^18^2

1^12^04^28^1     1^12^04^48^-1     1^12^04^58^-2     1^12^04^68^-3     1^12^14^-18^3     1^12^14^08^2

1^12^14^18^1     1^12^14^38^-1     1^12^14^48^-2     1^12^24^-28^3     1^12^24^-18^2     1^12^24^08^1

1^12^24^28^-1     1^12^24^38^-2     1^12^34^-38^3     1^12^34^-28^2     1^12^34^-18^1     1^12^34^18^-1

1^12^44^-38^2     1^12^44^-28^1     1^12^44^08^-1     1^12^54^-38^1     1^22^-44^78^-1     1^22^-44^88^-2

1^22^-34^48^1     1^22^-34^68^-1     1^22^-34^78^-2     1^22^-34^88^-3     1^22^-24^28^2     1^22^-24^38^1

1^22^-24^58^-1     1^22^-24^68^-2     1^22^-24^78^-3     1^22^-14^18^2     1^22^-14^28^1     1^22^-14^48^-1

1^22^-14^58^-2     1^22^-14^68^-3     1^22^04^-18^3     1^22^04^08^2     1^22^04^18^1     1^22^04^38^-1

1^22^04^48^-2     1^22^14^-28^3     1^22^14^-18^2     1^22^14^08^1     1^22^14^28^-1     1^22^24^-38^3

1^22^24^-28^2     1^22^24^-18^1     1^32^-54^88^-2     1^32^-54^98^-3     1^32^-44^68^-1     1^32^-44^78^-2

1^32^-44^88^-3     1^32^-34^28^2     1^32^-34^38^1     1^32^-34^58^-1     1^32^-34^68^-2     1^32^-34^78^-3

1^32^-24^08^3     1^32^-24^18^2     1^32^-24^28^1     1^32^-24^48^-1     1^32^-14^-18^3     1^32^-14^08^2

1^32^-14^18^1     1^32^04^-28^3              

6lN=8, k=5/2

1^-32^44^58^-1     1^-32^54^28^1     1^-32^54^48^-1     1^-32^54^58^-2     1^-32^64^18^1     1^-32^64^38^-1

1^-32^74^-18^2     1^-32^74^08^1     1^-22^34^58^-1     1^-22^44^28^1     1^-22^44^48^-1     1^-22^44^58^-2

1^-22^54^18^1     1^-22^54^38^-1     1^-22^54^48^-2     1^-22^54^58^-3     1^-22^64^-18^2     1^-22^64^08^1

1^-22^64^28^-1     1^-22^74^-28^2     1^-22^74^-18^1     1^-22^84^-48^3     1^-12^14^68^-1     1^-12^24^38^1

1^-12^24^58^-1     1^-12^24^68^-2     1^-12^34^28^1     1^-12^34^48^-1     1^-12^34^58^-2     1^-12^34^68^-3

1^-12^44^08^2     1^-12^44^18^1     1^-12^44^38^-1     1^-12^44^48^-2     1^-12^44^58^-3     1^-12^54^-18^2

1^-12^54^08^1     1^-12^54^28^-1     1^-12^54^38^-2     1^-12^54^48^-3     1^-12^64^-38^3     1^-12^64^-28^2

1^-12^64^-18^1     1^-12^64^18^-1     1^-12^74^-48^3     1^-12^74^-38^2     1^-12^74^-28^1     1^-12^84^-58^3

1^02^04^68^-1     1^02^14^38^1     1^02^14^58^-1     1^02^14^68^-2     1^02^24^28^1     1^02^24^48^-1

1^02^24^58^-2     1^02^24^68^-3     1^02^34^08^2     1^02^34^18^1     1^02^34^38^-1     1^02^34^48^-2

1^02^34^58^-3     1^02^44^-18^2     1^02^44^08^1     1^02^44^28^-1     1^02^44^38^-2     1^02^44^48^-3

1^02^54^-38^3     1^02^54^-28^2     1^02^54^-18^1     1^02^54^18^-1     1^02^54^28^-2     1^02^64^-48^3

1^02^64^-38^2     1^02^64^-28^1     1^02^64^08^-1     1^02^74^-58^3     1^02^74^-48^2     1^02^74^-38^1

1^12^-24^78^-1     1^12^-14^48^1     1^12^-14^68^-1     1^12^-14^78^-2     1^12^04^38^1     1^12^04^58^-1

1^12^04^68^-2     1^12^04^78^-3     1^12^14^18^2     1^12^14^28^1     1^12^14^48^-1     1^12^14^58^-2

1^12^14^68^-3     1^12^24^08^2     1^12^24^18^1     1^12^24^38^-1     1^12^24^48^-2     1^12^24^58^-3

1^12^34^-28^3     1^12^34^-18^2     1^12^34^08^1     1^12^34^28^-1     1^12^44^-38^3     1^12^44^-28^2

1^12^44^-18^1     1^12^54^-48^3     1^22^-34^78^-1     1^22^-24^48^1     1^22^-24^68^-1     1^22^-24^78^-2

1^22^-14^38^1     1^22^-14^58^-1     1^22^-14^68^-2     1^22^-14^78^-3     1^22^04^18^2     1^22^04^28^1

1^22^04^48^-1     1^22^14^08^2     1^22^14^18^1     1^22^24^-28^3     1^32^-54^88^-1     1^32^-44^58^1

1^32^-44^78^-1     1^32^-44^88^-2     1^32^-34^48^1     1^32^-34^68^-1     1^32^-24^28^2     1^32^-24^38^1

6lN=8, k=3

1^-22^54^48^-1     1^-22^64^18^1     1^-12^34^58^-1     1^-12^44^28^1     1^-12^44^48^-1     1^-12^44^58^-2

1^-12^54^18^1     1^-12^54^38^-1     1^-12^64^-18^2     1^-12^64^08^1     1^02^24^58^-1     1^02^34^28^1

1^02^34^48^-1     1^02^34^58^-2     1^02^44^18^1     1^02^44^38^-1     1^02^44^48^-2     1^02^44^58^-3

1^02^54^-18^2     1^02^54^08^1     1^02^54^28^-1     1^02^64^-28^2     1^02^64^-18^1     1^02^74^-48^3

1^12^04^68^-1     1^12^14^38^1     1^12^14^58^-1     1^12^14^68^-2     1^12^24^28^1     1^12^24^48^-1

1^12^34^08^2     1^12^34^18^1     1^22^-14^68^-1     1^22^04^38^1        

6lN=8, k=7/2

1^02^44^48^-1     1^02^54^18^1              

6lN=9, k=1/2

1^03^09^1                 

6lN=9, k=1

1^-13^49^-1     1^03^09^2     1^03^19^1     1^03^39^-1     1^13^-19^2     1^13^09^1

1^23^-19^1                 

6lN=9, k=3/2

1^-23^79^-2     1^-13^39^1     1^-13^59^-1     1^03^19^2     1^03^29^1     1^03^49^-1

1^13^09^2     1^13^19^1     1^13^39^-1     1^23^-19^2     1^23^09^1     

6lN=9, k=2

1^-23^79^-1     1^-23^89^-2     1^-13^49^1     1^-13^69^-1     1^-13^79^-2     1^03^39^1

1^03^59^-1     1^03^69^-2     1^13^19^2     1^13^29^1     1^13^49^-1     1^23^09^2

1^23^19^1                 

6lN=9, k=5/2

1^-23^89^-1     1^-23^99^-2     1^-13^59^1     1^-13^79^-1     1^-13^89^-2     1^03^49^1

1^03^69^-1     1^03^79^-2     1^13^39^1     1^13^59^-1     1^23^19^2     

6lN=9, k=3

1^-13^89^-1     1^03^79^-1     1^03^89^-2     1^13^49^1        

6lN=9, k=7/2

1^03^89^-1                 

6lN=10, k=1/2

1^02^05^-110^2     1^02^05^010^1     1^02^05^210^-1           

6lN=10, k=1

1^-12^25^-110^2     1^-12^25^010^1     1^-12^25^110^0     1^-12^25^210^-1     1^02^05^010^2     1^02^05^110^1

1^02^15^-110^2     1^02^15^010^1     1^02^15^110^0     1^02^15^210^-1     1^12^05^-110^2     1^12^05^010^1

1^12^05^210^-1     1^22^-15^-110^2     1^22^-15^010^1     1^22^-15^110^0     1^22^-15^210^-1     

6lN=10, k=3/2

1^-12^25^010^2     1^-12^25^110^1     1^-12^25^210^0     1^02^15^010^2     1^02^15^110^1     1^02^15^210^0

1^02^25^-110^2     1^02^25^010^1     1^02^25^110^0     1^02^25^210^-1     1^12^05^010^2     1^12^05^110^1

1^12^15^-110^2     1^12^15^010^1     1^12^15^110^0     1^12^15^210^-1     1^22^-15^010^2     1^22^-15^110^1

1^22^-15^210^0     1^22^05^-110^2     1^22^05^010^1     1^22^05^210^-1        

6lN=10, k=2

1^02^25^110^1     1^02^25^210^0     1^12^15^010^2     1^12^15^110^1     1^12^15^210^0     1^22^05^010^2

1^22^05^110^1                 

6lN=11, k=3/2

1^011^3                 

6lN=11, k=2

1^011^4     1^111^3     1^311^1           

6lN=11, k=5/2

1^111^4     1^411^1              

6lN=13, k=1/2

1^013^1                 

6lN=13, k=1

1^113^1                 

6lN=14, k=1

1^02^07^-214^4     1^02^07^-114^3     1^02^07^014^2     1^02^07^314^-1     1^02^07^414^-2     

6lN=14, k=3/2

1^-22^47^014^1     1^-22^47^114^0     1^-22^47^214^-1     1^-12^27^214^0     1^-12^27^314^-1     1^-12^27^414^-2

1^-12^37^014^1     1^-12^37^114^0     1^-12^37^214^-1     1^02^07^014^3     1^02^17^-214^4     1^02^17^-114^3

1^02^17^014^2     1^02^17^214^0     1^02^17^314^-1     1^02^17^414^-2     1^02^27^014^1     1^02^27^114^0

1^02^27^214^-1     1^12^07^-214^4     1^12^07^-114^3     1^12^07^014^2     1^12^07^314^-1     1^12^07^414^-2

1^22^-17^-214^4     1^22^-17^-114^3     1^22^-17^014^2     1^22^07^-114^2     1^22^07^014^1     1^32^-17^-114^2

1^32^-17^014^1     1^32^-17^114^0     1^42^-27^-114^2     1^42^-27^014^1     1^42^-27^114^0     

6lN=14, k=2

1^02^17^014^3     1^02^17^314^0     1^02^37^014^1     1^02^37^114^0     1^12^07^014^3     1^32^07^014^1

6lN=15, k=1

1^03^05^-115^3     1^03^05^015^2     1^03^05^315^-1           

6lN=15, k=3/2

1^-13^35^015^1     1^-13^35^115^0     1^03^05^015^3     1^03^15^-115^3     1^03^15^015^2     1^03^15^215^0

1^03^15^315^-1     1^03^25^015^1     1^03^25^115^0     1^13^05^-115^3     1^13^05^015^2     1^13^05^315^-1

1^23^05^015^1     1^33^-15^015^1     1^33^-15^115^0           

6lN=15, k=2

1^03^15^315^0     1^03^35^115^0     1^13^05^015^3     1^33^05^015^1        

6lN=17, k=1

1^017^2                 

6lN=17, k=3/2

1^117^2     1^217^1              

6lN=19, k=1

1^019^2                 

6lN=19, k=3/2

1^119^2     1^219^1              

6lN=21, k=1

1^03^07^021^2                 

6lN=21, k=3/2

1^03^17^221^0     1^03^27^121^0     1^13^07^021^2     1^23^07^021^1        

6lN=27, k=1/2

1^03^09^027^1                 

6lN=27, k=1

1^03^09^127^1     1^03^19^027^1     1^03^29^-127^1           

6lN=27, k=3/2

1^03^09^227^1     1^03^19^127^1     1^03^29^027^1           

6lN=27, k=2

1^03^29^127^1                 




Each entry in Table <ref> is different from others. For instance, although the eta-quotient η(τ)^3, when considered as η(τ)^3η(2τ)^0, generates the one-dimensional space M_3/2(Γ_0(2),χ) where χΓ_0(3)→^× is the character of η(τ)^3η(2τ)^0, we list η(τ)^3 only in the group N=1, not N=2,k=3/2 to avoid repetition. As a comparison, Tables 2 and 3 of <cit.>, whose entries constitute a subset of Table <ref> above, contain repeated eta-quotients. For example, η(τ)^-1η(2τ)^2=η(τ)^-1η(2τ)^2η(4τ)^0 is listed both in the groups corresponding to N=2,k=1/2 and N=4,k=1/2 there.



One should notice, e.g., that the formal product 1^03^05^015^2 which represents η(15τ)^2 is in Table <ref> but 1^03^05^015^1 is not. This is because _ M_1/2(Γ_0(15),χ) can not be computed using Theorem <ref> (with t=0) for any χ but _ M_1(Γ_0(15),χ) can with χ being the character of η(15τ)^2 as one may directly see by comparing both sides of the first inequality of (<ref>). See also Table <ref> and Table <ref>. Nevertheless, we can prove that η(15τ) is weakly admissible, as the following proposition shows.



Let f be an eta-quotient (of integral exponents). If there is a positive integer n such that f^n is weakly admissible, then f is weakly admissible. As a consequence, The m-th root, where m∈Z1, of an eta-quotient ∏η(nτ)^r_n in Table <ref> is weakly admissible provided that m| r_n for all n.


Let N, k be the level and weight of f respectively. Let χΓ_0(N)→^× be the character of f. Then χ^n is the character[Here χ^n means the character on Γ_0(N) that sends (γ,ε) to χ(γ,ε)^n where γ∈Γ_0(N) and ε∈{±1}.] of f^n. First suppose f^n is weakly admissible of type I, which means f^n is holomorphic at all cusps and _ M_nk(Γ_0(N),χ^n)=1. Then obviously f is holomorphic at all cusps so f∈ M_k(Γ_0(N),χ). To prove _ M_k(Γ_0(N),χ)=1, let 0≠ g∈ M_k(Γ_0(N),χ) be arbitrary. Then g^n∈ M_nk(Γ_0(N),χ^n) and hence g^n=c· f^n for some c∈^×. It follows that g=c_1· f where c_1 is an n-th root of c since  is connected and f is holomorphic on . This proves that f is weakly admissible of type I. The case of type II is proved in a similar manner with spaces such as M_k(Γ_0(N),χ) replaced by S_k(Γ_0(N),χ).


It should be noted that if f^n is weakly admissible of type II, then we can only assert that f is weakly admissible; we can say nothing about whether f is of type I or II. For instance, consider f(τ)=η(τ), n=24.




 §.§ Generalized double coset operators

We recall the theory of generalized double coset operators developed in <cit.>. This is a generalization of Wohlfahrt's extension of Hecke operators; c.f. <cit.>. Let l, N be positive integers. Let χ be the character (<ref>) where D=2 and r_n are integers. Let (r_n')_n| N be another sequence of integers such that ∑_nr_n=∑_nr_n' and let χ' be the character (<ref>) with r_n replaced by r_n' (again D=2). We define a formal expression (c.f. <cit.>)

    T_lf(τ)=l^-k/2·∑_a| l(N,a)=1a^k∑_0≤ b<d(a,b,d)=1χ-Nb+axz-Ndy^-1χ'ayby-dzNx^-1f(aτ+b/d)

where k∈1/2, d=l/a and x,y,z are any integers (depending on a, b and d) such that (Nd,-Nb+ax)=1 and (-Nb+ax)y+Ndz=1. If the dependence on k, N, χ and χ' is important, then we also denote T_l by T_l;k,N,χ,χ' or T_l;χ,χ'.


For f being a meromorphic modular form of weight k∈1/2 for the group Γ_0(N) with character χ, T_lf is well-defined (i.e., independent of the choices of x,y,z) and is a meromorphic modular form of the same weight for the same group with character χ' provided that the following three conditions hold:

    l∑_n | NN/nr_n ≡∑_n | NN/nr_n' 24,
    ∑_n | Nnr_n ≡ l∑_n | Nnr_n' 24,
    
    l^2k'·∏_2 ∤ r_n-r_n'n  is a perfect square,

where k'=1/2∑_nr_n. Moreover, if the above conditions hold, then T_l maps M_k(Γ_0(N),χ) into M_k(Γ_0(N),χ') and S_k(Γ_0(N),χ) into S_k(Γ_0(N),χ') respectively.


This is a combination of the last assertion of <cit.>, <cit.> and <cit.>.



Let f_1=∏_nη(nτ)^r_n and f_2=∏_nη(nτ)^r'_n be weakly admissible eta-quotients of the same type (e.g., functions listed in Table <ref>) of the same weight k and the same level N. Let χ and χ' be the characters of f_1 and f_2 on Γ_0(N) respectively. Then for any l∈Z1 satisfying (<ref>), (<ref>) and (<ref>) (with k'=k), there exists a c_l∈ such that

    T_l;χ,χ'f_1=c_l· f_2.



An immediate consequence of Theorem <ref>.


This theorem is still valid in the following two cases:

  
  * f_1 is weakly admissible of type II and f_2 is weakly admissible of type I,
  
  * f_1 is a cusp form that is weakly admissible of type I and f_2 is weakly admissible of type II.



In the case N=1 (hence r_1=r'_1, k=1/2r_1 and χ=χ') the identity reads T_lη^r_1=c_l·η^r_1 where r_1=0,1,2,…24. This has been investigated in detail in <cit.> where c_l and (<ref>) are given in explicit forms.


We are mainly interested in the case f_1=f_2 in Theorem <ref>.


Let f(τ)=∏_nη(nτ)^r_n be a weakly admissible eta-quotient (e.g., a function listed in Table <ref>) of weight k=1/2∑_nr_n and level N. We define a submonoid of the multiplicative monoid of positive integers as follows:

    L_f={l∈Z1 l≡1m_f}    if  k∈,
    {l∈Z1 l≡1m_f and l is a perfect square}    if  k∈1/2+,

where 
    m_f=24/(24,∑_n(N/n)r_n,∑_nnr_n).

Then for any l∈ L_f we have T_lf=c_l· f for some c_l∈.


Elementary number theory shows that l∈ L_f if and only if (<ref>), (<ref>) and (<ref>) with r'_n=r_n and k'=k hold. Therefore the assertion follows from Theorem <ref> with f_1=f_2=f.



The monoid L_f and the integer m_f actually depend only on 𝐫=(r_n)_n| N, so we sometimes write L_𝐫 and m_𝐫 instead.

In another words, any weakly admissible eta-quotient f is an eigenform for the operators T_l indexed by the infinite monoid L_f. When k∈ and the character of f is induced by some Dirichlet character, then by a result of Gordon, Hughes and Newman (cf. <cit.>) we have m_f=1 and hence L_f=Z1. Moreover, in this case, T_l is proportional to the usual Hecke operator of index l. Consequently, the functions in Table <ref> are all Hecke eigenforms for appropriate infinite operator monoids in a generalized sense.

The identities T_lf=c_l· f become interesting only when we can express (<ref>) in a more explicit form, at least one where x,y,z do not appear. The following theorem gives such a formula which is key to the current topic. The reader may compare this with a previous result of Wohlfahrt <cit.>.


Let N∈Z1. For each n| N, let r_n be an integer and set 

    𝐫=(r_n)_n| N,   k'=1/2∑_n| Nr_n,   x_N=∑_n| Nnr_n,  =∏_n| N(Nn^-1)^r_n.

Let χ_𝐫Γ_0(N)→^× be the character of ∏_n| Nη(nτ)^r_n, i.e. (<ref>). Let f be a meromorphic modular form on Γ_0(N) of weight k∈ k'+2 with character χ_𝐫 and let l∈ L_𝐫. Then

  
  * we have the expansions (on τ>Y_0)
  
    f(τ)=∑_n∈x_N/24+c_f(n)q^n,   T_lf(τ)=∑_n∈x_N/24+c_T_lf(n)q^n,

  where Y_0 is some nonnegative number, c_f(n), c_T_lf(n)∈ are uniquely determined and T_l is the operator defined in (<ref>) with χ=χ'=χ_𝐫;
  
  * for n∈x_N/24+ we have
  
    c_T_lf(n)=l^-k/2∑_a| l, d=l/a(a,N)=1aa^k· c_f(ln/a^2)∑_0≤ b<d(a,b,d)=1-Nb(a,d)^2kbd(n/l-x_N/24)ψ_l,𝐫(a,b)

  where
  
    ψ_l,𝐫(a,b)=-k+δ/4(d-1)+(k+δ)(l-1)(N-1)/4    if  2∤ l,
    -k+δ/4(a-1)-N(k+δ)(1+δ_1)/4b    if  2| l, 2| N,
    
      1     if  2| l, 2∤ N,

  with δ=0 (δ=1 respectively) if  takes the form 2^α·(4m+1) (2^α·(4m+3) respectively), α, m∈Z0, δ_1=1 if 2| l, 4| N, k∈1/2+, v_2()≡12 and δ_1=0 otherwise.


A few words of caution are in order. Note that a and -Nb(a,d) refer to Kronecker-Jacobi symbols; they are not fractions. Also note that in (<ref>) we have set c_f(ln/a^2)=0 if ln/a^2∉x_N/24+. In addition, k can be replaced by k' in the definition of ψ_l,𝐫(a,b) and if f is the eta-quotient ∏_n| Nη(nτ)^r_n itself, which is the case we will use below, then k=k'. Finally, Y_0=0 if f is holomorphic on .


The formula (<ref>) is equivalent to

    T_lf(τ)=l^-k/2∑_a| l, d=l/a(a,N)=1aa^k∑_0≤ b<d(a,b,d)=1-Nb(a,d)^2k-x_N/24bdψ_l,𝐫(a,b)· f(aτ+b/d)

which can be considered as an explicit form of (<ref>) with χ=χ'.


We begin with the proof of (<ref>). Set

    R:=χ-Nb+axz-Ndyχ'ayby-dzNx=χ_𝐫(-Nb+ax)ay+Nz(-Nb+ax)(by-dz)+xz-Nady+Ny-Nd(by-dz)+xy

in (<ref>). (Theorem <ref> and the assumption l∈ L_𝐫 ensure that T_l is well-defined.) Since ad=l and

    (-Nb+ax)y+Ndz=1

we have

    R=χ_𝐫a-N(l-1)zb-(l-1)xz-N(l-1)yd-(l-1)xy.

To derive (<ref>) from (<ref>), we need only to prove

    R^-1=a-Nb(a,d)^2k-x_N/24bdψ_l,𝐫(a,b)

for which we need the Petersson's formula of χ_η:

    χ_η(abcd,ε)=ε·dc1/24((a-2d)c-bd(c^2-1)+(3d-3)c)   if 2 ∤ c, 
    ε·cd1/24((a-2d)c-bd(c^2-1)+3d-3)   if 2 | c.

For a proof of (<ref>) see <cit.>. Since r_n∈ we have

    R=∏_n| Nχ_η^r_na-N(l-1)zn(b-(l-1)xz)-N/n(l-1)yd-(l-1)xy.

The strategy[The N=1 case has been successfully worked out in the proof of <cit.>. The strategy here is essentially the one there.] is to insert (<ref>) into (<ref>) and then eliminate x, y, z using properties of Kronecker-Jacobi symbols. Inserting (<ref>) into (<ref>) gives

    R=∏_n| N2∤N/n(l-1)yd-(l-1)xyNn^-1(l-1)y^r_n·∏_n| N2|N/n(l-1)y-Nn^-1(l-1)yd-(l-1)xy^r_n·x_N/24bd
    ·∏_n| N2∤N/n(l-1)yr_n/8(d-1-(l-1)xy)(-Nn^-1)(l-1)y·∏_n| N2|N/n(l-1)yr_n/8(d-1-(l-1)xy)

where we have used the fact

    24|(l-1)∑_n| Nnr_n,  24|(l-1)∑_n| NNn^-1r_n

which are consequences of the assumption l∈ L_𝐫. Now the proof splits into three cases.

The case 2∤ l. We must have 2|N/n(l-1)y for any n| N. Therefore

    R=x_N/24bd∏_n| N-Nn^-1(l-1)yd-(l-1)xy^r_n∏_n| Nr_n/8(d-1-(l-1)xy).

Note that if k'∈1/2+ then l is an odd square and hence

    -Nn^-1(l-1)yd-(l-1)xy=Nn^-1d-(l-1)xy·-(l-1)yd-(l-1)xy=Nn^-1d-(l-1)xy·yd.

Inserting this into (<ref>) we find that

    R=x_N/24bd·d-(l-1)xyyd^2k'k'/4(d-1-(l-1)xy).

If k'∈, the above expression is still valid. By (<ref>) if k'∈1/2+ we have

    yd=-Nb+axd=-Nb(a,d).

For the last equality, see <cit.>. If k'∈, then trivially yd^2k'=-Nb(a,d)^2k'. For the calculation of d-(l-1)xy, we set

    '=∏_p|, 2∤ v_p()p.

There are four cases to consider. For the first case, '≡14, we have

    d-(l-1)xy='d-(l-1)xy=d-(l-1)xy'=a'=a.

The third equality above follows from

    d-(l-1)xyp=d-lxy+xyp=xyp=ap,   p|'

since by (<ref>) we have axy≡1p and lxy≡ dp. Moreover, we have

    -k'(l-1)xy/4=-k'(l-1)(N-1)/4.

(When k'∈, l≡34 and 2∤ N, (<ref>) does not necessarily hold, but we can always choose x,y,z such that 2| y for which (<ref>) holds. We always make such a choice.) Inserting (<ref>), (<ref>) and (<ref>) into (<ref>) and replacing k' by k we obtain (<ref>) as required. For the other three cases '≡34, '≡28 and '≡68, we proceed similarly and obtain (<ref>) likewise. However, if 2|' there is an extra difficulty: a factor 2ad-(l-1)axy=21+(l-1)N(dz-by) appears and we should prove it equals 1. This is immediate if l≡14 since 2|' implies 2| N. Now suppose l≡34. Assume by contradiction that 4∤ N; we have N≡24. By the first relation of (<ref>) we have 4|∑_nnr_n from which 2|∑_2∤ nr_n. Since 2|' we have

    v_2()=∑_n| Nv_2(N/n)·r_n=∑_2∤ nr_n≡12

which contradicts 2|∑_2∤ nr_n. Therefore, 4| N and hence 21+(l-1)N(dz-by)=1.

The case 2| l, 2∤ N. By (<ref>) if 2| d then 2∤ y and if 2∤ d then we can choose x,y,z such that 2∤ y. (If (x,y,z) is a solution to (<ref>) with 2| y, then (x,y+Nd,z-(-Nb+ax)) is another solution with 2∤ y+Nd.) We always make such a choice. Thus (<ref>) becomes

    R=x_N/24bd∏_n| Nd-(l-1)xyNn^-1(l-1)y^r_n∏_n| Nr_n/8(d-1-(l-1)xy)(-Nn^-1)(l-1)y.

Since l≡1m_𝐫 we have 2∤ m_𝐫. By the definition of m_𝐫 (c.f. Remark <ref>) we have

    8|∑_nnr_n,   8|∑_nNn^-1r_n.

It follows that k'=1/2∑_nr_n∈ and

    R=x_N/24bd∏_n| Nd-(l-1)xyNn^-1(l-1)y^r_n.

Note that

    d-(l-1)xyNn^-1(l-1)y=d-(l-1)xyNn^-1·d-(l-1)xy(l-1)y=d-(l-1)xyNn^-1·d(l-1)y

where the last equality follows from the fact (l-1)y is odd and positive. Inserting this and 2k'≡02 into (<ref>) we obtain

    R=x_N/24bd·d-(l-1)xy=x_N/24bd·a

where the last equality follows just as in the deduction of (<ref>). We thus have arrived at (<ref>).

The case 2| l, 2| N. As in the last case, we have 8|∑_nnr_n and 8|∑_nNn^-1r_n. Moreover, y must be odd by (<ref>). For any n| N with 2∤N/n(l-1)y, that is, with 2∤N/n, we have

    d-(l-1)xyNn^-1(l-1)y=Nn^-1(l-1)yd-(l-1)xy·-1/8(d-1-(l-1)xy)·(Nn^-1(l-1)y-1).

Choosing a solution (x,y,z) to (<ref>) with y<0 and inserting the above identity into (<ref>), we find that

    R=x_N/24bd·∏_n| N-Nn^-1(l-1)yd-(l-1)xy^r_n·∏_n| N2∤N/n-r_n/8(d-1-(l-1)xy)(-Nn^-1(l-1)y-1)
    ·∏_n| N2∤N/nr_n/8(d-1-(l-1)xy)(-Nn^-1)(l-1)y·∏_n| N2|N/nr_n/8(d-1-(l-1)xy)

We deal with the factor ∏_n| N-Nn^-1(l-1)yd-(l-1)xy^r_n as in previous cases and thus simplify the expression for R as

    R=x_N/24bd·d-(l-1)xy·d-(l-1)y^2k'·k'/4(d-1-(l-1)xy)·(-(l-1)y).

There are six subcases: (c.f. (<ref>) for ')

  
  * 2| l, 4| N, k'∈ and 2∤'.
  
  * 2| l, 4| N, k'∈ and 2|'.
  
  * 2| l, 4| N, k'∈1/2+ and 2∤'.
  
  * 2| l, 4| N, k'∈1/2+ and 2|'. (Only in this subcase we have δ_1=1.)
  
  * 2| l, N≡24 and 2∤'. (We must have k'∈ in this subcase.)
  
  * 2| l, N≡24 and 2|'. (We must have k'∈ in this subcase.)

We will only give the proof of the more difficult subcases (b) and (d) and omit the other four since the strategies are the same. For (b), we first prove 8| N. Assume by contradiction that 8∤ N; then 2^2∥ N. It follows that

    v_2()≡∑_n| Nv_2(n)=1v_2(N/n)·r_n=∑_n| Nv_2(n)=1r_n≡12.

On the other hand, (<ref>) still holds in this case, and hence

    ∑_n| Nv_2(n)=0r_n≡∑_n| Nv_2(n)=2r_n≡02.

Therefore ∑_n| Nr_n≡12, that is, k'∈1/2+, which contradicts the assumption in (b). Hence 8| N. Note that both '/2 and d-(l-1)xy are odd and positive, from which we deduce that

    d-(l-1)xy   ='d-(l-1)xy
       =2d-(l-1)xyd-(l-1)xy'/21/4(d-1-(l-1)xy)·'/2-1/2
       =2ad-(l-1)axya'1/4(d-1-(l-1)xy)·'/2-1/2
       =21+(l-1)N(dz-by)a1/4(d-1-(l-1)xy)·'/2-1/2
       =a1/4(a-1)·'/2-1/2.

The last equality holds since by (<ref>) and 8| N we have axy≡14 and hence

    (a-1)-(d-1-(l-1)xy)≡ a-d+(l-1)a=(a^2-1)d≡04.

An argument similar to the above deduction shows that

    k'/4(d-1-(l-1)xy)·(-(l-1)y)=k'/4(a-1)(l-1).

Inserting this and (<ref>) into (<ref>) and noting that a is odd, we obtain (<ref>) in the subcase (b). For the subcase (d), l is a square. Since (N,a)=1 and ad=l we have v_2(d)=v_2(l)≡02 and hence 4| d. Thus if we let d'=d/2^v_2(d) then

    d-(l-1)y=dy=d'y=y-1/4·d'-1/2·yd.

Combining this and

    -Nb+axdyd=(-Nb+ax)yd=1-Ndzd=1

we find that

    d-(l-1)y=y-1/4·d'-1/2·-Nb+axd=y-1/4·d'-1/2·-Nb(a,d).

The calculation of d-(l-1)xy is similar to that in above cases (e.g. (<ref>) or (<ref>)) and the result is

    d-(l-1)xy=21+Nba1/4(l-1)(a-1)·'/2-1/2.

For the factor k'/4(d-1-(l-1)xy)·(-(l-1)y) in (<ref>), note that l-d=(a-1)d≡08, 2∤ y and hence

    k'/4(d-1-(l-1)xy)·(-(l-1)y)   =k'/4(l-1)^2(1-xy)(-y)
       =k'/4(1-xy)(-y)=k'/4(x-y).

Inserting this equality, (<ref>) and (<ref>) into (<ref>) we obtain

    R=a-Nb(a,d)x_N/24bd21+Nb
    ·1/4(l-1)(a-1)·'/2-1/2y-1/4·d'-1/2+k'/4(x-y).

Now we simplify the factor y-1/4·d'-1/2+k'/4(x-y) in above. By (<ref>) and the facts 2| d, 4| N we find that (-Nb+ax)y≡18. This, together with 2∤ y, implies x≡ ay+Nab8. It follows that

    y-1/4·d'-1/2+k'/4(x-y)   =-2k'/4(y-1)·d'-1/2+k'/4(ay-y+Nab)
       =k'/4Nabk'/4y(a-d')k'/4(d'-1).

Since ad'=l/2^v_2(d)=l/2^v_2(l) is a square, 2∤ a and 2∤ d' we have

    a-d'≡08.

In addition, since k'∈1/2+, 2∤ a and 4| N,

    k'/4Nab=1/8Nab=1/8Nb.

Inserting (<ref>) and (<ref>) into (<ref>), then combining (<ref>) and noting that 1/8Nb21+Nb=1, we arrive at (<ref>) as required. This concludes the proof of (<ref>).

Finally, we begin to prove Theorem <ref>. For Theorem <ref>(a), elementary Fourier analysis shows that

    f(τ)=∑_n∈1/24c_f(n)q^n,   T_lf(τ)=∑_n∈1/24c_T_lf(n)q^n,

which converge on τ>Y_0 for some Y_0≥0 since f(τ+24)=f(τ) and T_lf(τ+24)=T_lf(τ). By modularity f|_k1101=χ_𝐫1101· f=x_N/24· f and T_lf|_k1101=χ_𝐫1101· T_lf=x_N/24· T_lf. In terms of Fourier coefficients,

    c_f(n)n=c_f(n)x_N/24,   c_T_lf(n)n=c_T_lf(n)x_N/24.

It follows that c_f(n)=0 unless n∈x_N/24+ and so is c_T_lf(n). Theorem <ref>(b), that is, (<ref>), follows immediately from (<ref>) whose details are omitted.


In the important case x_N≡024 we have c_f(n)=0 and c_T_lf(n)=0 unless n∈. Therefore, (<ref>) becomes

    c_T_lf(n)=l^-k/2∑_a| l, a^2| ln(a,N)=1, d=l/aaa^k· c_f(ln/a^2)∑_0≤ b<d(a,b,d)=1-Nb(a,d)^2kbdn/lψ_l,𝐫(a,b)

for n∈. From this some elegant formulas for special cases follow:

We keep the notations and assumptions of Theorem <ref> and assume that x_N≡024. Then

    c_T_lf(0)=c_f(0)· l^-k/2∑_a| l, d=l/a(a,N)=1aa^k∑_0≤ b<d(a,b,d)=1-Nb(a,d)^2kψ_l,𝐫(a,b).

Moreover, if for any prime p not dividing N we have p^2∤ l (e.g. l is square-free or (l)|(N)), then

    c_T_lf(1)=c_f(l)· l^-k/2∑_0≤ b<lψ_l,𝐫(1,b).

Finally, if l=p^α where p is a prime not dividing N and α∈Z1, then

    c_T_p^αf(1)=p^-α k/2∑_0≤β≤[α/2]p^β p^kβ· c_f(p^α-2β)∑_0≤ b<p^α-β(p^β,b)=1-Nbp^β^2kb/p^βψ_l,𝐫(p^β,b).



The sum over b in the above formulas, (<ref>) or (<ref>) is either a partial sum of a geometric sequence or a Gauss sum according to whether k is integral or half-integral. It is not hard to derive an explicit formula for this sum but we exclude this task here. For the N=1 case the reader may consult <cit.> and <cit.> for such formulas.




 §.§ Identities involving Fourier coefficients of weakly admissible eta-quotients


The main theorem of Section <ref> is the following:


Let f(τ)=∏_n| Nη(nτ)^r_n=∑_nc_f(n)q^n be a weakly admissible eta-quotient (e.g., a function listed in Table <ref>) where N is the level. We keep the notations in (<ref>) with k'=k. Then for any l∈ L_f (c.f. Corollary <ref>) there exists a unique c_l∈ such that

    l^-k/2∑_a| l, d=l/a(a,N)=1aa^k· c_f(ln/a^2)∑_0≤ b<d(a,b,d)=1-Nb(a,d)^2kbd(n/l-x_N/24)ψ_l,𝐫(a,b)=c_l· c_f(n)

holds for any n∈x_N/24+.
Moreover, the number c_l is equal to the left-hand side of the above identity with n replaced by x_N/24.


By Corollary <ref> T_lf=c_l· f and hence c_T_lf(n)=c_l· c_f(n) for any n∈x_N/24+. Inserting (<ref>) into this identity gives the desired formula. Letting n=x_N/24 and noting that c_f(x_N/24)=1 we obtain the formula for c_l.


The identities (<ref>) have been verified numerically for l≤121, n-x_N/24≤40 and f being any function listed in Table <ref> by a SageMath program. See Appendix <ref> for the code. The rest of this section is devoted to useful consequences of Theorem <ref> and concrete examples.



  §.§.§ Closed formulas for Fourier coefficients of eta-quotients.
 Let f(τ)=∏_n| Nη(nτ)^r_n be a weakly admissible eta-quotient (e.g., a function listed in Table <ref>) where N is the level. We keep the notations in (<ref>) with k'=k. Here let us consider the situation x_N=0 where we have

    f(τ)=∏_n| N∏_m∈Z1(1-q^nm)^r_n=∑_n∈Z0c_f(n)q^n,  τ∈.

We have given a series expression of c_f(n) involving Dedekind sums for N=4, r_1=2, r_2=7, r_4=-4 in Example <ref>. This example is of course also listed in Table <ref>. In fact, there exist very simple closed formulas for c_f(n) only involving Kronecker-Jacobi symbols at least for n square-free—thanks to Theorem <ref>.


Let the notations and assumptions be as above. Let l∈ L_f (c.f. Corollary <ref>) satisfy

    p∤ N p^2∤ l,  for any prime  p.

(e.g. l is square-free or (l)|(N).) We require that at least one of l and N is odd. Then

    c_f(l)=
    -r_1·∑_a| l, (a,N)=1aa^k-1    if  2| l, 2∤ N,
    
    -r_1·∑_a| l, (a,N)=1aa^k-1(k+δ)(l-d)/4    if  2∤ l,

where d=l/a, =∏_n| N(Nn^-1)^r_n and δ=0 (δ=1 respectively) if  takes the form 2^α·(4m+1) (2^α·(4m+3) respectively). In particular, if (l)|(N), then c_f(l)=-r_1.


A combination of (<ref>) and (<ref>) (with n=1) gives c_f(l)·∑_0≤ b<lψ_l,𝐫(1,b)=l^k/2c_l· c_f(1). By the last assertion of Theorem <ref> we have

    c_l=l^-k/2∑_a| l, (a,N)=1aa^k· c_f(0)∑_0≤ b<dψ_l,𝐫(a,b).

Since c_f(0)=1 and c_f(1)=-r_1 we find that

    c_f(l)·∑_0≤ b<lψ_l,𝐫(1,b)=-r_1·∑_a| l, (a,N)=1aa^k·∑_0≤ b<dψ_l,𝐫(a,b).

The assumption at least one of l and N is odd ensures that ∑_0≤ b<lψ_l,𝐫(1,b)≠0. Therefore the desired formulas follow by expanding ∑_0≤ b<lψ_l,𝐫(1,b) and ∑_0≤ b<dψ_l,𝐫(a,b) in (<ref>).


The formulas for c_f(l) in the above theorem seem like coefficients of Eisenstein series. This is actually the case since x_N=0 implies f is not a cusp form. Since f is weakly admissible, then it must be of type I, that is, _ M_k(Γ_0(N),χ_𝐫)=1. Therefore f is in the Eisenstein space.


In the case of half-integral weights, Theorem <ref> can be used only for those l with (l)|(N). However, there is no half-integral weight eta-quotient f in Table <ref> such that c_f(l) (l>1) can be computed using Theorem <ref>. Nevertheless, it is still worthwhile to include the case of half-integral weights in Theorem <ref> since there may exist weakly admissible eta-quotients not listed in Table <ref> whose coefficients c_f(l) can be computed using this theorem.



Let us consider an example of level 12 which is not listed in Table <ref>. The reader may use SageMath programs described in Appendix <ref> to produce this eta-quotient and check the assertions given here. Set

    f(τ)=η(τ)^1η(2τ)^-1η(3τ)^-1η(4τ)^1η(6τ)^4η(12τ)^-2=∑_n∈Z0c_f(n)q^n.

Let (x_c)_c| 12 be as in Proposition <ref>. Hence ÷_a/cf=x_c/24 for c| 12 and a coprime to c; c.f. (<ref>). We have

    [  x_1;  x_2;  x_3;  x_4;  x_6; x_12 ]=
    [ 12  6  4  3  2  1;  3  6  1  3  2  1;  4  2 12  1  6  3;  3  6  1 12  2  4;  1  2  3  1  6  3;  1  2  3  4  6 12 ][  r_1;  r_2;  r_3;  r_4;  r_6; r_12 ]=
    [ 11;  5;  9;  8; 15;  0 ],

from which we see that f is a holomorphic eta-quotient. Moreover, f is admissible of type I. Thus Theorem <ref> can be applied to f. Note that m_f in Corollary <ref> is equal to 24 and hence

    L_f={l∈Z1 l≡124}.

Therefore (a,12)=1 for any a| l where l∈ L_f. Moreover,

    k=1,  =2^9·3^3=square·6,  δ=1.

It follows from Theorem <ref> that

    c_f(l)=-∑_a| la6=-∏_p| l(1+p6)

where 1≤ l≡124 is square-free. Consequently, for such l, -c_f(l) is either zero or a power of 2. Moreover, for any α∈Z1 there exist infinitely many l such that c_f(l)=-2^α.





  §.§.§ Multiplicativity of weakly admissible eta-quotients.
 The Fourier coefficients c_f(n) of an eta-quotient f in Martin's list <cit.> satisfy c_f(nm)c_f(1)=c_f(n)c_f(m) whenever n, m are coprime positive integers. This is the reason they are called multiplicative eta-quotients. The proper generalization to weakly admissible eta-quotients is stated in the following theorem.


Let us keep the notations and assumptions of Theorem <ref>. Then the map L_f→ that sends l to c_l is a multiplicative function, that is, we have c_l_1l_2=c_l_1· c_l_2 for any l_1,l_2∈ L_f with[The gcd (l_1,l_2) is the usual greatest common divisor in the monoid of positive integers, not in the monoid L_f. These two concepts are different. For instance in the monoid L={l∈Z1 l≡124}, (5·29,5·53)=5 in the usual sense, but the unique common divisor of 5·29 and 5·53 in L is 1.] (l_1,l_2)=1.






This theorem is an immediate consequence of the following general fact which in the special case N=1 is due to Wohlfahrt <cit.>.


Let N∈Z1 and χ_1, χ_2, χ_3 be complex linear characters on the double cover Γ_0(N). Let l_1,l_2∈Z1 such that χ_1 and χ_2 are 100l_1-compatible, χ_2 and χ_3 are 100l_2-compatible. Then χ_1 and χ_3 are 100l_1l_2-compatible. Moreover, if (l_1,l_2)=1 then

    T_l_2;χ_2,χ_3∘ T_l_1;χ_1,χ_2f=T_l_1l_2;χ_1,χ_3f

for any meromorphic modular form f on Γ_0(N) of weight k∈1/2 and with character χ_1.

For the concept of compatibility of multiplier systems, see <cit.> for details. We say χ and χ' are 100l-compatible if for any γ∈100l^-1Γ_0(N)100l∩Γ_0(N) we have χ(100lγ100l^-1)=χ'(γ).

First let us prove χ_1 and χ_3 are 100l_1l_2-compatible. Let g∈100l_1l_2^-1Γ_0(N)100l_1l_2∩Γ_0(N) be arbitrary. We have

    χ_1(100l_1l_2g100l_1l_2^-1)   =χ_1(100l_1100l_2g100l_2^-1100l_1^-1)
       =χ_2(100l_2g100l_2^-1)
       =χ_3(g).

In the second equality we have used the assumption χ_1 and χ_2 are 100l_1-compatible and in the third equality the assumption χ_2 and χ_3 are 100l_2-compatible. This proves the 100l_1l_2-compatibility of χ_1 and χ_3.

Assume that (l_1,l_2)=1. To prove (<ref>) we need an equivalent definition of T_lf via double coset actions. For the proof of this equivalence see <cit.>. Write

    Γ_0(N)100l_iΓ_0(N)=_γ∈ A_iΓ_0(N)γ

for i=1,2. Then

    T_l_1;χ_1,χ_2f=∑_γ∈ A_1(χ_1|_l_1χ_2)^-1(γ)· f|_kγ,

where χ_1|_l_1χ_2(h)=χ_1(h_1)χ_2(h_2) if we write h=h_1100l_1h_2 with h_1,h_2∈Γ_0(N). The compatibility ensures that χ_1|_l_1χ_2(h) is well-defined. Therefore

    T_l_2;χ_2,χ_3∘ T_l_1;χ_1,χ_2f=∑_γ_1∈ A_1,γ_2∈ A_2(χ_2|_l_2χ_3)^-1(γ_2)(χ_1|_l_1χ_2)^-1(γ_1)· f|_kγ_1γ_2.

By <cit.> with m=1 we can choose the following A_i:

    A_i={ab0d a,d>0, 0≤ b<d, ad=l_i, (N,a)=1, (a,b,d)=1}.

It follows from the assumption (l_1,l_2)=1 and elementary number theory that γ_1γ_2∈Γ_0(N)100l_1l_2Γ_0(N) if γ_1∈ A_1 and γ_2∈ A_2. Therefore the map

    σ A_1× A_2   →Γ_0(N)\Γ_0(N)100l_1l_2Γ_0(N)
    
    (γ_1,γ_2)    ↦Γ_0(N)·γ_1γ_2

is well-defined. By the theory of classical Hecke algebras, or more precisely, by Theorem 2.8.1(2) and Eq. (2.7.2) of <cit.>, σ is a surjection and each inverse image σ^-1(x) has the same cardinality. Since σ^-1(Γ_0(N)100l_1l_2)={(100l_1,100l_2)} is a singleton, σ is a bijection which means the range of summation in the right-hand side of (<ref>) is essentially a set of representatives of the coset space Γ_0(N)\Γ_0(N)100l_1l_2Γ_0(N). Now we can define a map, which is key to the proof,

    ξΓ_0(N)100l_1l_2Γ_0(N)   →
    
    x   ↦χ_1|_l_1χ_2(xγ_2^-1)·χ_2|_l_2χ_3(γ_2)

where (γ_1,γ_2) is the uniquely determined pair in A_1× A_2 such that Γ_0(N)x=Γ_0(N)γ_1γ_2. It is not hard to see that

    ξ(gxh)=χ_1(g)ξ(x)χ_3(h),   g,h∈Γ_0(N), x∈Γ_0(N)100l_1l_2Γ_0(N),  ξ100l_1l_2=1

and that

    ξ(γ_1γ_2)=χ_1|_l_1χ_2(γ_1)χ_2|_l_2χ_3(γ_2).

Since the map x↦χ_1|_l_1l_2χ_3(x) also satisfies (<ref>), we have ξ=χ_1|_l_1l_2χ_3. Inserting this and (<ref>) into (<ref>) gives

    T_l_2;χ_2,χ_3∘ T_l_1;χ_1,χ_2f=∑_γ_1∈ A_1,γ_2∈ A_2ξ^-1(γ_1γ_2)· f|_kγ_1γ_2=∑_γ_1∈ A_1,γ_2∈ A_2(χ_1|_l_1l_2χ_3)^-1(γ_1γ_2)· f|_kγ_1γ_2.

Since σ is a bijection, we obtain (<ref>) from the above equalities.

Finally we prove Theorem <ref>. Let f be as in this theorem. Then T_l_1f=c_l_1· f, T_l_2f=c_l_2· f and T_l_1l_2f=c_l_1l_2· f by Corollary <ref>. Since (l_1,l_2)=1 we have

    c_l_1l_2· f=T_l_1l_2f=T_l_2(T_l_1f)=T_l_2(c_l_1· f)=c_l_1c_l_2· f

where we have used Proposition <ref> in the second equality. Since f is not identically zero we find c_l_1l_2=c_l_1· c_l_2 as required.


Before looking at examples, let us simplify some notations. Set x_N'=x_N/24 and

    S_l,𝐫(a)=∑_0≤ b<d(a,b,d)=1-Nb(a,d)^2k-(l-1)x_N'b/aψ_l,𝐫(a,b).

Then according to Theorem <ref> we have

    c_l=l^-k/2∑_a| l, (a,N)=1aa^k· c_f(lx_N'/a^2)S_l,𝐫(a),   l∈ L_f.

We recall that here and below f must be a weakly admissible eta-quotient.

If x_N'=0, that is, x_N=∑_n| Nnr_n=0, then c_f(0)=1 and hence Theorem <ref> gives

    ∑_a| l_1, (a,N)=1aa^k· S_l_1,𝐫(a)·∑_a| l_2, (a,N)=1aa^k· S_l_2,𝐫(a)=∑_a| l_1l_2, (a,N)=1aa^k· S_l_1l_2,𝐫(a)

for l_1,l_2∈ L_f with (l_1,l_2)=1.



If x_N'∈Z0 and (l,x_N')=1, then c_f(lx_N'/a^2)=0 unless lx_N'/a^2∈ by Theorem <ref>(a). Therefore we have

    ∑_a^2| l_1, (a,N)=1aa^k· c_f(l_1x_N'/a^2)S_l_1,𝐫(a)·∑_a^2| l_2, (a,N)=1aa^k· c_f(l_2x_N'/a^2)S_l_2,𝐫(a)
    
    =∑_a^2| l_1l_2, (a,N)=1aa^k· c_f(l_1l_2x_N'/a^2)S_l_1l_2,𝐫(a)

if l_1,l_2∈ L_f with (l_1,l_2)=1, (l_1,x_N')=1 and (l_2,x_N')=1. If in addition (<ref>) holds for l=l_1,l_2, then the above identity becomes

    c_f(l_1x_N')S_l_1,𝐫(1)· c_f(l_2x_N')S_l_2,𝐫(1)=c_f(l_1l_2x_N')S_l_1l_2,𝐫(1).

From this we obtain a property of c_f(n) very closed to the multiplicativity property. (We exclude the half-integral case below for simplicity.)




Let f(τ)=∏_n| Nη(nτ)^r_n=∑_nc_f(n)q^n be a weakly admissible eta-quotient (e.g., a function listed in Table <ref> or Example <ref>) where N is the level. We keep the notations in (<ref>) with k'=k. Assume that x_N'=x_N/24∈Z0 and k∈. Let l_1,l_2∈ L_f satisfy (l_1,l_2)=1, (l_1,x_N')=1 and (l_2,x_N')=1. We require (<ref>) holds for l=l_1 and l_2. Moreover, if N≡02 and N(k+δ)≡24, then we require l_1≡ l_2≡12. (See Theorem <ref>(b) for the definition of δ.) Then

    c_f(l_1x_N')· c_f(l_2x_N')=ε· c_f(l_1l_2x_N'),

where ε=u_l_1,𝐫· u_l_2,𝐫· u_l_1l_2,𝐫 with

    u_l,𝐫=
    
    1,     if   2| l,  2∤ N  or 4| N(k+δ)
    
    1,     if   2∤ l,  4| N(k+δ)(l-1)
    
    -1,     if   2∤ l,  4∤ N(k+δ)(l-1).



According to (<ref>) we need to prove

    S_l_1l_2,𝐫(1)=u_l_1,𝐫· u_l_2,𝐫· u_l_1l_2,𝐫· S_l_1,𝐫(1)· S_l_2,𝐫(1).

By the definition of ψ_l,𝐫 in Theorem <ref>, we have, since k∈,

    S_l,𝐫(1)=
    
    l,     if   2| l,  2∤ N  or 4| N(k+δ)
    
    0,     if   2| l,  2| N and   N(k+δ)≡24
    
    l,     if   2∤ l,  4| N(k+δ)(l-1)
    
    -l,     if   2∤ l,  4∤ N(k+δ)(l-1).

From this (<ref>) follows. Substituting (<ref>) in (<ref>) we obtain

    c_f(l_1x_N')· c_f(l_2x_N')S_l_1,𝐫(1)S_l_2,𝐫(1)=ε· c_f(l_1l_2x_N')S_l_1,𝐫(1)S_l_2,𝐫(1).

By the assumption we have S_l_1,𝐫(1) and S_l_2,𝐫(1) are nonzero and hence the required relation follows.


If x_N'=0, then the above proposition holds trivially. We believe there are many weakly admissible eta-quotients with x_N'∈Z1 to which we can apply the above proposition. The most well-known example is the discriminant function Δ(τ)=η(τ)^24, which in our terminology is an admissible eta-quotient of type II. The only example in Table <ref> is the one presented below.


Set f=η(3τ)^2η(9τ)^-1η(27τ)=∑_n∈Z1c_f(n)q^n. This is an admissible eta-quotient of type I. We have

    N=27,   k=1,   x_N=24,  =3^5,  δ=1.

Since (c.f. (<ref>))

    [  x_1;  x_3;  x_9; x_27 ]=
    [ 27  9  3  1;  3  9  3  1;  1  3  9  3;  1  3  9 27 ][  r_1;  r_3;  r_9; r_27 ]=
    [ 16; 16;  0; 24 ],

we have m_f=3 (c.f. Corollary <ref>) and hence L_f={3n+1 n∈Z0}. It follows from Proposition <ref> that c_f(l_1)c_f(l_2)=c_f(l_1l_2) where l_1,l_2≡13 are square-free positive integers with (l_1,l_2)=1.



Let us give an example of admissible eta-quotient of type II. Set

    f=η(τ)η(2τ)η(3τ)η(6τ)^3=∑_n∈Z1c_f(n)q^n.

We have

    N=6,   k=3,   x_N=24,  =2^3·3^2,  δ=0.

Since

    [ x_1; x_2; x_3; x_6 ]=
    [ 6 3 2 1; 3 6 1 2; 2 1 6 3; 1 2 3 6 ][ r_1; r_2; r_3; r_6 ]=
    [ 14; 16; 18; 24 ],

we have m_f=12 and hence L_f={12n+1 n∈Z0}. By Theorem <ref>, _ M_3(Γ_0(6),χ)=2 where χΓ_0(6)→^× is the character of f. (Of course this descends to a character of the matrix group Γ_0(6) since k∈.) Recall that the number of Eisenstein series in M_3(Γ_0(6),χ) is denoted by n_0 in (<ref>). Since χ(γ_a/cT^w_a/cγ_a/c^-1)=1 if and only if x_c≡024 where c| 6 and (a,c)=1, among the four cusps in Γ_0(6)\ there is only one Eisenstein series which is defined at the cusp ∞. Therefore n_0=1. Since f is a cusp form, we find that it is admissible of type II. It follows from Proposition <ref> (applied to this f) that c_f(l_1)c_f(l_2)=c_f(l_1l_2) whenever l_1,l_2≡112 are square-free positive integers with (l_1,l_2)=1. It is worthwhile to notice that for other pair (l_1,l_2), the multiplicativity does not necessarily hold. For instance, c_f(2)=-1, c_f(3)=-2 but c_f(6)=4.




  §.§.§ An example of half-integral weight
 The examples presented above are all of integral weights. To conclude this section we supplement an example of half-integral weight where we show how the Gauss sum ∑_b in (<ref>) can be simplified. Let

    f(τ)=η(τ)^-7η(2τ)^17η(4τ)^-3=∑_n∈5/8+Z0c_f(n)q^n.

The statistics of f are

    N=4,   k=7/2,   x_1=3,  ,x_2=24,   x_4=15,  =2^31,  δ=0,   m_f=8.

Thus L_f={l^2 l≡12}. By Theorem <ref> we have _ M_7/2(Γ_0(4),χ)=2 where χΓ_0(4)→^× is the character of f and by Definition <ref> the Eisenstein series can only be given at the cusp 1/2. Thus, f is admissible of type II and Theorems <ref>, <ref> can be applied.

Now we calculate c_l^2 for l^2∈ L_f (c.f. Theorem <ref>). We have, by definition,

    c_l^2=l^-7/2∑_a| l^2a2a^7/2c_f(5l^2/8a^2)-7l^2/8a+7/8∑_0≤ b<l^2/a(a,b,l^2/a)=1-b(a,l^2/a)-5(l^2-1)b/8a.

Note that c_f(5l^2/8a^2)=0 unless 5l^2/8a^2∈5/8+, that is, unless a| l since a, l are odd. It follows that for such a we have (a,l^2/a)=a,

    t:=5(l^2-1)·(a,l^2/a)/8a=5(l^2-1)/8

is an integer (depending on a and l) and

    ∑_0≤ b<l^2/a(a,b,l^2/a)=1-b(a,l^2/a)-5(l^2-1)b/8a   =∑_0≤ b<l^2/a-ba-tb/a
       =l^2/a^2∑_0≤ b<abatb/a

is, up to a simple factor, a Gauss sum associated with the character ·(a,l^2/a). To express the values of these Gauss sums, we let, for any odd positive integer m,

    (m)    = ∏_p| mv_p(m)≡02p         (m)    = ∏_p| mv_p(m)≡12p 
    (m)    = (m)(m)        (m)    = (m)^2(m) 
    (m)    = m/(m)        (m)   =m/(m)

where p denotes a prime.
By <cit.>, if (a)∤ t then ∑_0≤ b<abatb/a=0; if (a)| t then

    ∑_0≤ b<abatb/a
    =ε_aa/√((a))t/(a)(a)∏_p |(a)(p-1-pt/(a)p^2),

where t/(a)(a), t/(a)p are Jacobi symbols and ε_a=1 (ε_a= respectively) if a≡14 (a≡34 respectively). Inserting (<ref>) and (<ref>) into (<ref>) and noting that -7l^2/8a+7/8=a-1/8, a2ε_aa-1/8=1 we obtain the following formula.

For f(τ)=η(τ)^-7η(2τ)^17η(4τ)^-3=∑_n∈5/8+Z0c_f(n)q^n and 1≤ l≡12 we have

    c_l^2=l^-3/2∑_a| l(a)| ta^5/2/√((a))∏_p |(a)(p-1-pt/(a)p^2)t/(a)(a)c_f(5l^2/8a^2)

where t=5(l^2-1)/8. In particular, if l is square-free, then

    c_l^2=l^-3/2∑_a| la^2·tac_f(5l^2/8a^2).



Since a^5/2/√((a))∈ and c_f(n)∈, we have l^3/2· c_l^2∈.


According to Theorem <ref> (applied to (<ref>)) we have c_l_1^2c_l_2^2=c_l_1^2l_2^2 whenever l_1,l_2 are coprime odd integers. In the special case l_1 and l_2 are square-free, this identity reads

    ∑_a| l_1a^2·5(l_1^2-1)/8ac_f(5l_1^2/8a^2)·∑_a| l_2a^2·5(l_2^2-1)/8ac_f(5l_2^2/8a^2)
    
    =∑_a| l_1l_2a^2·5(l_1^2l_2^2-1)/8ac_f(5l_1^2l_2^2/8a^2).

From this we find the genuine multiplicativity property of (<ref>) as stated below. The reader may compare this to the integral-weight counterpart—Proposition <ref>.


Let f(τ)=η(τ)^-7η(2τ)^17η(4τ)^-3=∑_n∈5/8+Z0c_f(n)q^n. Let l_1,l_2 be coprime odd square-free integers. Then

    c_f(5l_1^2/8)c_f(5l_2^2/8)=c_f(5l_1^2l_2^2/8).



First suppose l_1 and l_2 are primes. By (<ref>) we have

    (c_f(5l_1^2/8)+l_1^2·5(l_1^2-1)/8l_1)·(c_f(5l_2^2/8)+l_2^2·5(l_2^2-1)/8l_2)
    
    =c_f(5l_1^2l_2^2/8)+l_2^2·5(l_1^2l_2^2-1)/8l_2c_f(5l_1^2/8)+l_1^2·5(l_1^2l_2^2-1)/8l_1c_f(5l_2^2/8)+l_1^2l_2^2·5(l_1^2l_2^2-1)/8l_1l_2.

It is not hard to prove, by the periodicity and multiplicativity of Jacobi symbols, that

    5(l_1^2l_2^2-1)/8l_1   =5(l_1^2-1)/8l_1,
    5(l_1^2l_2^2-1)/8l_2   =5(l_2^2-1)/8l_2,
    5(l_1^2l_2^2-1)/8l_1l_2   =5(l_1^2-1)/8l_1·5(l_2^2-1)/8l_2.

Inserting these into (<ref>) gives the desired identity in the case l_1 and l_2 are primes.

Second, suppose l_1 is a prime and l_2 is arbitrary. Again by (<ref>) we have

    (c_f(5l_1^2/8)+l_1^2·5(l_1^2-1)/8l_1)·∑_a| l_2a^2·5(l_2^2-1)/8ac_f(5l_2^2/8a^2)
    
    =∑_a| l_2a^2·5(l_1^2l_2^2-1)/8ac_f(5l_1^2l_2^2/8a^2)+∑_a| l_2l_1^2a^2·5(l_1^2l_2^2-1)/8l_1ac_f(5l_2^2/8a^2).

Since for a| l_2,

    5(l_1^2-1)/8l_1·5(l_2^2-1)/8a=5(l_1^2l_2^2-1)/8l_1a,

we obtain from (<ref>) that

    c_f(5l_1^2/8)·∑_a| l_2a^2·5(l_2^2-1)/8ac_f(5l_2^2/8a^2)   =∑_a| l_2a^2·5(l_1^2l_2^2-1)/8ac_f(5l_1^2l_2^2/8a^2)
       =∑_a| l_2a^2·5(l_2^2-1)/8ac_f(5l_1^2l_2^2/8a^2).

The desired identities for l_1 prime and l_2 arbitrary now follows from (<ref>), the already proven case and the induction on the number of prime divisors of l_2.

Finally, if l_1 and l_2 are arbitrary, then, by the already proven cases, both sides of the desired identity are equal to ∏_p| l_1l_2c_f(5p^2/8) and hence the identity holds which concludes the proof.


We have discussed consequences of Theorem <ref> applied to (<ref>). The above exploration is also valid for any admissible eta-quotient of half-integral weight. (Of course the integral weight case is simpler to deal with.) In addition, for (<ref>) or any admissible eta-quotient of half-integral weight, we can as well simplify the left-hand side of (<ref>) as we have just done for c_l^2. This will give more interesting identities about c_f(n). We will not include more details about this due to the length of the paper.




§ MISCELLANEOUS OBSERVATIONS AND OPEN QUESTIONS


More general multiplier systems. The dimension formulas stated in Theorem <ref> concern multiplier systems induced by eta-quotients of fractional exponents. As a comparison, most of the formulas that have appeared in the literature concern multiplier systems induced by Dirichlet characters (in the case of integral weights) or Dirichlet characters times the multiplier system of a power of η(τ)^-2η(2τ)^5η(4τ)^-2 (in the case of half-integral weights). It is meaningful to deduce an explicit formula for _ M_k(Γ_0(N),χ_1χ_2) where k∈, χ_1 is induced by an eta-quotient of fractional exponents and χ_2 by a Dirichlet character.

Do generalized double coset operators exist for rational weights? Theorem <ref> let us know for what numbers l does the expression T_lf make sense. The multiplier systems involved are required to be induced by eta-quotients of integral exponents. The question is: if the multiplier systems are induced by eta-quotients of fractional exponents, is there any nontrivial T_l (i.e. l>1)? All identities presented in Section <ref> are based on Theorem <ref>. Thus if we can find any T_l with l>1 in the case of fractional exponents, then, taking into account of Theorem <ref>, we would obtain identities involving c_f(n) for infinitely many eta-quotients f of fractional exponents. However, no such operator has been found up to now.

L-functions and Euler products. According to Theorem <ref>, it is natural to associate an L-function L(f,s)=∑_l∈ L_fc_l/l^s with a weakly admissible eta-quotient f. This association is different from the usual one ∑_nc_f(n)/n^s. The identities presented in Section <ref> can be rephrased as properties of this L(f,s). For instance, it has an Euler product. It is interesting to investigate L(f,s), e.g., the functional equations, the corresponding Weil's theorem, etc.

Listing all weakly admissible eta-quotients. Open question: to find out all weakly admissible eta-quotients (if there exist only finitely many). Another open question: for each weakly admissible eta-quotient, to deduce identities like the one in Proposition <ref> from (<ref>).




§ USAGE OF SAGEMATH CODE


The SageMath code can be obtained from the repository <cit.>. It is an ipynb file which should be opened in a Sage Jupyter notebook. The code in the first two cells should be run after the file is opened. The first cell, which actually is the one in <cit.>, contains Python/Sage functions dealing with eta-quotients among which we need those concerning the characters and Fourier coefficients of eta-quotients of fractional exponents. The second cell then contains Python/Sage functions on computing dimensions, on order-character relations, on checking identities and on generating tables in this paper.



elsarticle-num



 Paper Title: Non degenerate Birkhoff functions.


§ ABSTRACT   It is known after the works of Mahler, Kleinbock, Margulis, Sprindžuk that very well approximated numbers on a manifold form a zero measure set, assuming non-degeneracy conditions.  These non-degeneracy conditions are,
  in many applications, difficult to check. We propose here a setting in which these are automatic. Rather than functions, we consider correspondences which have a better behaviour.

Interpretable correlator Transformer for image-like quantum matter data
    Annabelle Bohrdt
    August 5, 2024
=======================================================================



 


§ INTRODUCTION

In 1942, Siegel proved that a biholomorphic germ

    f:(,0) → (,0),  z ↦ z +o(|z|)

can be linearised, assuming an arithmetic condition on the coefficient
 <cit.>. Ever since, the field of Diophantine
approximation has been of relevance for dynamical systems and this led
Arnold, Margulis and Pyartli and others to reconsider the classical works
on Diophantine approximation and extend their scope <cit.>. Nowadays, the work of Kleinbock and Margulis finds application
in KAM theory in order to prove that in certain situations the quasiperiodic
solutions make up a set of positive measure in phase space <cit.>.
To apply these results, one needs to assume certain non-degeneracy conditions, introduced in the works of Mahler, Sprindžuk, Margulis and Kleinbock <cit.>. To our knowledge the weakest of these conditions was introduced by Kleinbock:
 Let U be an open set in ^d. A C^k-map B:U →^e is called KMS-non-degenerate (for Kleinbock-Margulis-Sprindžuk) in an affine space  if:

 
  * B(U) ⊂.
 
  * For some l ≤ k the partial derivatives at any point of U of order l span the vector space associated to . 


It is not difficult to see that real analytic maps are always KMS non-degenerate (see <ref>) and this will constitute our starting point. A simple example of a KMS-degenerate function is given by the flat function:

    →, x ↦ e^-1/x^2

 
 
 Assuming the KMS non-degeneracy condition, one may estimate the preimage under the map B of points which are badly approximated by the integer lattice ^e ⊂^e in various ways. For instance, under the KMS non-degeneracy condition, Kleinbock-Margulis and Kleinbock proved that the preimages of points satisfying an adequate Diophantine condition form a full measure set (<cit.>, <cit.>, see also <cit.>). 
  
Unfortunately the verification of non-degeneracy conditions for the frequency map of a dynamical system often rely on tedious computations and are notoriously difficult to establish. For instance, it took  many years before Féjoz and Herman proposed a proof of the non-degeneracy condition for the N-body problem initiated by Arnol'd in the sixties (<cit.>, see also <cit.>). In this paper, we will show that for the type of normal forms introduced by the authors in <cit.> the KMS-non degeneracy conditions are essentially automatic. So let us now explain what type of generalisation of the usual setting is needed by considering an instructive example.


 §.§ The Poincaré correspondences




In Les Méthodes Nouvelles de la Mécanique Céleste <cit.>. 
Poincaré considered the series


    s=1+x/1+y+x^2/1+2y+…=∑_k=0^∞x^k/1+k y

as a prototype of the kind of series that appear in perturbation theory (see also the paper <cit.> of Sauzin for the resurgence properties). In fact, similar expressions play a role in many areas of mathematics, for example in the theory of q-analogs of classical series (logarithm, exponential, hypergeometric), see for instance <cit.>.

Assume now that we want to solve the equation

    0=f(x,y):=-y+x/1+y+x^2/1+2 y+ +x^3/1+3 y+… .


If we truncate the series at x^n we obtain the rational functions

    f_n(x,y)=-y+∑_k=1^n x^k/1+ky =P_n(x,y)/D_n(y)

which determines a polynomial  P_n(x,y) in the variables x and y:

    P_1(x,y)=-(1+y)y+x,   P_2(x,y)=-(1+y)(1+2y)y+(1+2y)x+x^2, …


    D_1(y)=1+y,   D_2(y)=(1+y)(1+2y),    D_3(y)=(1+y)(1+2y)(1+3y),…

These polynomials define algebraic curves

    V_n:={ (x,y) ∈^2:P_n(x,y)=0 }⊂×

of degree (n+1) in y and degree n in x. These curves exhibit a peculiar bending behaviour caused by the poles, located on horizontal lines, defined by the denominators D_n(y) that were cleared out. 


  
Such a polynomial relation defines y as a multivalued function of x, sometimes called a correspondence. In a neighbourhood of x=0,y=0, the n-th curve V_n is the graph of a holomorphic function b_n that can be expanded in power series:

    [                  b_3(x)                       =    x+2x^5-14x^6+64x^7+…;                  b_4(x)                       = x+x^4-3x^5+7x^6-19x^7+…;                  b_5(x)                       =  x+x^4-2x^5+x^6+12x^7+…;                  b_6(x)                       =  x+x^4-2x^5+2x^6+5x^7+…;                         ]

It is not difficult to see that b_n+1(x)=b_n(x)+o(x^n), so these series converge coefficientwise to a formal power series solution b(x) ∈[[x]] to the equation f(x,y)=0

    b(x)=x+x^4-2x^5+2x^6+6x^7-35x^8+86x^9+…

We call it the Birkhoff expansion of the series f and its truncation are called the Birkhoff polynomials of the series.

 
 

 



The domain of definition of the b_n shrinks as n →∞ due to the peculiar winding, which provides a geometrical reason for the divergence of the series b(x). This is an illustration of a general divergence theorem for such expressions, that we intend to publish elsewhere <cit.>. On the other hand, the pictures strongly suggest that the part of curves in the part y>0 nicely converge to the graph of a C^∞-function B.


We may construct such a C^∞ function as follows: start with closed disks

    D_ρ:={x ∈: |x| ≤ρ}, D_η:={y ∈: |y| ≤η},

where ρ<1  and remove from the disk D_η small disks
D_k:={ y ∈: |y+1/k| < r_k } around the poles y=-1/k.
The distance between two nearby poles being 1/k(k+1),
 if we choose r_k=k^-, >2, the complement of the disks D_k contains the origin. We have in the complement of D_k an obvious estimate:
 
    |x^k/1+ky|≤ k^-1|x^k| ≤ k^-1ρ^k

So for  ρ <1, we see that
 the sequence obtained by restricting the rational functions f_n(x,y) to the closed set 
 
    X_∞ :=D_ρ×( D_η∖∪_k D_k )

converges in the C^0-topology to a limit function, that we denote in the same way: 

    f: X_∞





 §.§ Whitney differentiability

The closed set X_∞ does contain 0, but it is not an interior point as the poles accumulate at the origin. However, the formal power series b  that solves the equation f(x,b(x))=0 is the Taylor expansion of a function, C^∞ in a sense originally discovered by Borel and now called Whitney differentiability <cit.> Let us explain this property.  

If X ⊂^k is a locally closed subset, then a function 

    g:X →
 is called Whitney differentiable if  there exists a map

    L:X →  Hom(^k,)

such that

    g(y)=g(x)+L(x)(y-x)+o( x-y) .

The main difference with usual derivatives is the requirement that the variables x,y play a more
symmetric role. One defines similarly higher derivatives and Whitney C^∞ functions. Holomorphicity in the Whitney setting is defined by requiring that the map L is -linear
(^d=^2d) and in that case the function was called monogenic by Borel.

The Whitney extension theorem states that Whitney differentiable functions are restrictions of usual differentiable functions and the Whitney derivative commutes with restriction <cit.>. Fefferman showed that the extension operator can be chosen to be bounded linear for any finite level of differentiability <cit.>. In particular, the space of functions with bounded Whitney derivatives up to order k form a Banach space, like for ordinary differentiability.

Let us now show that our function f are indeed Whitney differentiable on X_∞. The proof is elementary but is worthwhile understanding. We will keep the notation f_n for the restriction of f_n to X_∞.  

We first create a second sequence X'_n by starting on a slightly bigger disc D_ρ+, ρ+<1, and by removing from D_η disks D'_k ⊂ D_k of radius smaller than r_k, say for instance r_k/2, for k ≤ n,  so that
X_n ⊂ X'_n. 
The Cauchy inequalities give an estimate for the derivatives of a holomorphic function on X'_n restricted to X_n:

    sup_(x,y) ∈ X_n|_̣x^i_̣y^j g(x,y) | ≤2^j i! j!/r_k^j^isup_(x,y) ∈ X'_n| g(x,y) |


Using the estimate on the complement of  D'_k:
 
    |x^k/1+ky|<2 k^-1|x^k|<2 k^-1(ρ+)^k,

and the Cauchy estimates, one easily deduces  that,  for any k ≥ 0, the sequence (f_n) is a Cauchy sequence in the C^k-Whitney holomorphic sense  and therefore the sum function f is Whitney C^∞
in the holomorphic sense.

 By the Whitney extension theorem, the function f is the restriction to X_∞ of a standard C^∞ function

    F:^2 →

The set F=0 serves as C^∞-version of the limit correspondence defined by f=0.
As _̣y F(0)=_̣y f(0)=1 ≠ 0, the usual C^∞ implicit function
theorem shows that in a neighbourhood of the origin the set  F=0 is the
graph of a C^∞ function B, that we call a C^∞
Birkhoff-function attached to f(x,y). Clearly, the Taylor-series of B is b.

Somehow we proved more than expected, as the function B is now also defined for y<0. This may look surprising, if we think of the way the real parts of the curves V_n have multiple components which stretch along the horizontal lines
near the x-axis. The resolution of this paradox is that by removing the tubes
D_ρ× D_k, we removed the bendings of the curve at the same time.



 §.§ Content of the paper
 
If one wants to apply the Kleinbock-Margulis theorem, one should be able to define and establish the KMS non-degeneracy condition for the correspondence before going to the Birkhoff series and doing the Whitney extension. The purpose of this paper is to show that this is indeed the case.

That the Birkhoff series b does not see any of the bendings may explain
heuristically why establishing the KMS non-degeneracy condition based on considerations of b alone turns out to be very difficult. Our approach consists of working as much as possible with
the correspondences lying on the product space, rather than with
the implicit functions they define. So most of the paper concerns the definition and analysis of non-degeneracy conditions for correspondences.

In the last part, we explain the application of the results of this paper to
the Herman conjecture. We also hope that our result, elementary as it may look, is of interest independently of the context in which it originally emerged,
and that the deep connection between number theory and dynamical systems can
be further strengthened, in the spirit of Siegel, Arnold and Margulis. 
 
 
 

§ STATEMENT OF THE THEOREM

 

 §.§ Birkhoff functions, series and polynomials
  
 
  We consider the space ^N=^d ×^e with coordinates x=(x_1,…,x_d), y=(y_1,…,y_e) and projections:
  
 
    ^N [rd]^q [ld]_p    
    ^d       ^e

We then consider a decreasing chain X=(X_n) of locally closed subsets of ^N:

    X_0 ⊃ X_1 ⊃ X_2 ⊃…⊃ X_∞ :=∩_n X_n .

and assume that the sets X_n are equal to the closure of their interior:
X_n^∘=X_n.
 We also assume that X_n is of the form
 
    X_n=×_n

 where ⊂^d is a neighbourhood of the origin.

Such sequences arise often in constructions where
X_n+1 is obtained from X_n by removing some open subset.
Note that the closed set X_∞ in general could have empty interior.
We also assume to be given a formal series (in multi-index notations):

    f(x,y)=∑_I ∈^d  a_I(y)x^I, a_I ∈{ y }

where { y } denotes the ring of convergent power series in y=(y_1,…,y_e).
We identify the coefficients of the series with functions of z=(x,y) depending only on y. We use the multi-index notation and consider the Whitney C^-norms:

    a ^_U= sup_ z ∈ U ∩ X_∞ |^̣_y a(z)| , ∈^e

where the partial derivatives are taken only in the y-direction as the functions involved as coefficients of the series only depend on y.
  We say the series ∑_I ∈^d  a_I(y)x^I is X-convergent if
  there is a neighbourhood U of the origin
  such that for any ∈^e, the power series
  
    ∑_I ∈^d  a_I^_U x^I ∈[[x]]

 is convergent in p(U).

An  X-convergent series in U ∩ X_∞ defines a Whitney C^∞ function that we denote by the same name:

    f:U ∩ X_∞→, (x,y) ↦∑_I ∈^d  a_I(y)x^I

Note that for (x_0,y_0) ∈ U ∩ X_∞, the map-germ

    (^d,x_0) →, x ↦ f(x,y_0)

is an ordinary holomorphic map-germ.

 
We use the notation:

    V(f):={(x,y) ∈ U ∩ X_∞ |  f(x,y)=0}

where f=(f_1,…,f_k) are X-convergent series
and we let (V(f),0) be the germ of this set at the origin. 

A Birkhoff function-germ associated to  X-convergent series f=(f_1,…,f_e), where _̣y f(0) has maximal rank, is a C^∞-function germ: 

    B:(^d,0) → (^e,0)
 whose graph (denoted _B) satisfies: 

    (_B ∩ X_∞,0)= (V(f),0).


Birkhoff polynomials and Birkhoff series are defined in a similar way by considering infinitesimal  and formal neighbourhoods instead of germs at the origin.

Note that the Taylor expansion of a Birkhoff function B at the origin is the Birkhoff series of f. Indeed by the Whitney-Fefferman theorem (<cit.>), we may choose for any k>0 and any neighbourhood U of the origin a (real) continuous linear extension operator[We consider independently the real and imaginary parts when doing the extension.] :

    T:C^k(U ∩ X_∞,) → C^k(U,)

The series

    F(x,y)=∑_I ∈^d  T(a_I(y))x^I

defines a C^∞ function-germ.
 We have
  
    F(x,b_k(x))=o( x ^k)
   
  where b_k is the degree k Birkhoff polynomial of f.   The manifolds { y=B(x) }, { F=0 } and {y=b_k(x) } are therefore tangent at order k at the origin. Hence the degree k Taylor polynomial of the function B is the degree k Birkhoff polynomial.
  
 

 §.§ Non-degeneracy theorem
  
   
 If f=∑_I ∈^d  a_I(y)x^I is an X-convergent series then we may, by Whitney's theorem, extend f to a C^∞ function-germ F. By the implicit function theorem, the zero locus of F is then the graph of a Birkhoff function-germ. So Birkhoff functions always exists. The aim of this paper is to show that the Birkhoff function can be chosen to be KMS non-degenerate: 

 
 Assume that the formal series:

    f(x,y)=∑_I ∈^d  a_I(y)x^I

is X-convergent and that the matrix _̣y f(0,0):=(_̣y_j f_i(0,0)) is of maximal rank. Then
the series f admits KMS non-degenerate Birkhoff function germs, i.e., there exists Whitney C^∞ germs:

    B:(^d,0) → (^e,0)  with  (_B ∩ X_∞,0)= (V(f),0).




It is useful to work over the field of complex numbers but of course our constructions commute with complex conjugation, in particular if the a_I are real analytic and X invariant under complex conjugation then the function B can be chosen to be real, i.e.:

    B(x̅,y̅)=B(x,y)

Our construction is going to be explicit and not just an existence theorem.
 





§ MAPS, NON-DEGENERACY AND THE KLEINBOCK LEMMA





 §.§ The span

For a subset X ⊂^e, we denote by [ X ] the span of X, defined as the smallest affine subspace  containing X . 


Clearly, if X ⊂ Y, then [ X ] ⊂ [ Y ]. 
For a germ (X,p) of a set we define its span as

    [ X,p ]:=⋂_U ∈(p) [ U ∩ X ] ,

where U runs over the set (p) of open neighbourhoods of the point p.
We always can find a sufficiently small representative U for the germ (X,p),
so that  [ X,p ]=[ U ].


For an open set U ⊂^d and map g:U ^e we also define its
span as [ g(U) ], and we similarly define the span of a map germ
g:(^d,) → (^e,)̱ at  as the affine space

    [ g, ] :=∩_U [ g(U),]̱ ,

where the intersection runs over all representatives of the germ g.


 Let g: U ^e be an analytic map and U() the
  connected component of U containing . Then g(U(α)) is
  contained in the span of the germ of g at :

    g(U()) ⊂ [ g,  ].

Consequently,

    [ g(U()) ]=[ g,  ] .






 §.§ The jet-span of a map

Given a multi-index I ∈^d, we denote by

    ^̣Ig():=∂^I g/∂ x^I() ∈^e

the vector of I-th derivative of g at . We will also use the notation

    | I|=i_1+…+i_d, I=(i_1,…,i_d).


We define the m-jet span of a Whitney C^∞ map 

    g:X →^e

at a point ∈ X as the affine space:

    [ g,] _m:=[{^̣I g(): I ∈^d, |I| ≤ m} ]


The m-jet spans  form a increasing chain, so become
stable, at some minimal index τ=τ(g,):

    [ g, ]_τ=[ g,  ]_τ+1=…

We call  τ(g,) the torsion index of the map g at the point . 
The condition [ g, ]_τ=[ g, ] corresponds to the KMS non-degeneracy condition.  

As an illustration, consider the germ

    g:(,0) → (^4,0):x ↦ (1+x,x^3,x^100,x+x^3+x^100)=(y_1,…,y_4)

We have

    [g,0]_0 ⊂ [ g,0]_1 =[ g,0]_2 ⊂ [ g,0]_3= [ g,0]_4= … =[ g,0]_99⊂ [ g,0]_100=[ g,0]_101=….

The span [ g,0 ] is the affine hyperplane [ g,0]_100={ y_4=y_1+y_2+y_3+1 } and the map-germ has torsion index τ(g,0)=100 at the origin. It is KMS non-degenerate, as any holomorphic map.


A flat C^∞ map-germ like

    g:(,0) → (,0), x ↦{ e^-1/x^2 if Re  x> 0 
    
    0  otherwise.

has torsion index equal to zero at the origin but it is KMS degenerate. Indeed,
  the span of this map germ is [ g,0] =, but [ g,0]_n={ 0 } for any n therefore:
  
    [g,0]_n ≠ [ g,0 ]

   



 §.§ A lemma of Kleinbock


The following observation due to Kleinbock was formulated originally in
the real analytic setting:

  If g:(^d,)  (^e,)̱ is a holomorphic map germ, then it  is KMS non degenerate.


For simplicity assume that =̱0 so that the span is a vector space.
Let g:U ^e be a connected representative of the germ (g,)
The sequence of spans [ g, ]_n generated by the vectors ^̣Ig(), |I| ≤ n stabilises at level τ. Clearly [ g, ]_τ⊂ [ g(U) ]. To show the other inclusion, let ℓ be an affine linear form vanishing on [ g, ]_τ. Then the Taylor series of ℓ∘ g at
 vanishes identically and therefore the holomorphic function ℓ∘ g vanishes on all of U. As this holds for all ℓ defining [ g,  ]_τ, we find [ g(U)] ⊂ [ g,  ]_τ, hence we have equality.


As we saw previously, in the C^∞ case, we may very well have a proper inclusion:

    ⋃_k [ g, ]_k ⊂ [g,]  but ⋃_k [ g, ]_k ≠ [g,].







§ CORRESPONDENCES AND NON-DEGENERACY




 §.§ Maps and correspondences

Many naturally appearing expressions like x^2+y^2=1 define y only
as a multivalued function y=√(1-x^2). Especially in the complex
domain, insisting on working with a monovalent function is cumbersome and
not very enlightening. Therefore, in algebraic geometry one is used to
working with correspondences, which provide a convenient extension of
the usual strict concept of function. The Poincaré example shows that this point of view is suited to the type of objects we are considering.  So our purpose will be to adapt the previous notions defined for maps to correspondences.







In this paper we will use the following terminology:


  A complex analytic variety V ⊂^N=^d ×^e is said to
  define a regular correspondence at ø=(,)̱, if the following
  conditions
  are verified:

  1) the set V is smooth of dimension d at ø.

  2) Under the projection onto the first factor, the tangent space T_øV
 maps isomorphically onto ^d.


Of course, by the implicit function theorem, we then can find a local
parametrisation g:(^d,)  (^e,)̱, x ↦ y=g(x)
so that (x, g(x)) ∈ V, or (Γ_g,ø) =(V,ø), where Γ_g
denotes the graph of g.




 §.§ The relative span

For a subset X ⊂^N:=^d ×^e, we denote by X the relative span of X, defined as the smallest affine subspace of the form ^d × F containing X where F ⊂^e is an affine subspace.
Clearly, one has

    X = q^-1(q([X])) ,

where q:^d ×^e ^e is the projection on the second factor.

Of course, the notions of span of a set and of a map germ are related:

Let Γ_g ⊂^d ×^e=^N denote the graph of
g:U ^e and let 
q:^N →^e be the projection onto the second factor. Then


    [ g(U)] =q(Γ_g)
  


 Let ℓ:  ^e → be an affine function vanishing on
[ g(U)] then

    ℓ':  ^d ×^e →, (x,y) ↦ℓ(y)

is an affine function vanishing on _g and conversely.
 


We define the relative span at a point ø∈ X
as the affine space

    X,ø  =∩_U  X ∩ U,ø ,

where the intersection runs over all neighbourhoods of ø∈^N and get an analog of Lemma <ref>:

 Let V ⊂^N be an analytic set and V(ø) the
connected component of V that contains the point ø∈ V. Then V(ø)
is contained in the span of the germ (V,ø):

    V(ø) ⊂ V, ø

and consequently:

    V(ø)= V, ø .




Let ℓ be an affine function vanishing on V, ø. As a holomorphic
function vanishing on the germ (V,ø), it vanishes on a neighbourhood of V
containing ø. By the identity theorem of analytic functions, ℓ vanishes
on the connected component V(ø).




 §.§ The relative jet-span of an analytic set

For a germ of an analytic set (V, ø) we define the n-jet span 

    V, ø_n

 as the smallest affine subspace of the form ^d × F that contains the n-th infinitesimal
neighbourhood of the fibre at ø∈ V of the projection ^N →^d. It is alternatively defined by the kernel of the affine linear functions contained in the ideal I+M_ø,x^n+1 which depend only on the y-variables. Here y_1,…,y_e denote coordinates on ^e, I ⊂_^N,p
is the ideal defining (V,ø) and M_ø,x⊂_^N,ø the maximal
ideal of holomorphic functions vanishing at x=, ø=(,)̱. 

Clearly, one has


    V, ø_1  ⊂ V, ø_2 ⊂ V, ø_3 ⊂…,

so this chain stabilises for a certain index τ:=τ(V,p), that we call the  torsion index of V at p:

    V, ø_τ  =  V, ø_τ+1=…

 
 For any holomorphic map-germ g:(^d,) → (^e,)̱ we have

    q(_g,ω_m)=[g,]_m

for ø=(,)̱ and any m ∈.


Write _g,_m={ℓ=0 } where 

    ℓ:^d ×^e →^r, (x,y) ↦ℓ(y)
 is some affine map depending only on the y-variables. By the Taylor formula, we have

    ℓ(g(x))=∑_ |I| ≤ m^̣Ig()/|I|!(x-)^I+o( x ^m)

and therefore the conditions ℓ(g(x))=o( x ^m) and ^̣Ig()=0 for |I| ≤ m are equivalent.














 §.§ A variant of the Kleinbock lemma



The following result is the analog of the Kleinbock lemma for analytic spaces:

 
 Let (V,ø) ⊂ (^N,0) be a germ of an analytic space with torsion index τ at ø then:

    V,ø =  V,ø_τ




  Clearly, V,ø_τ⊂ V,ø. For the converse, note that we have
  ∩_n M_p^n =(0), so that a function that vanishes on all infinitesimal
  neighbourhoods of ø vanishes on (V,ø), from which we obtain the reverse inclusion.
 







 §.§ n-Correspondences and the finite determinacy lemma

We now consider truncated sums as we did in the introduction for the Poincaré example:
  A correspondence V is called an n-correspondence at ω,
if it is defined by d equations of the form   

    f(x,y) =∑_|I| ≤ n, I ∈^d a_I(y)x^I ∈{ y}[x]

 
 


 



 
 The following lemma is a refinement of Lemma <ref>:

 
  
  If V is an n-correspondence at ω, then 
  
    τ(V,ω) ≤ n

  that is: V,ω_n = V, ω.


  
Consider an affine function ℓ vanishing on
  V,ω_n  depending only on the y-variables.
 Let v ∈^d be an arbitrary vector and consider the function
  
    :(,0) → (^e,0), t ↦ f(+tv,y)

  By definition of the n-span,  the polynomial ℓ∘ in the variable t has a zero of multiplicity ≥ n+1 at the origin:
  
    (ℓ∘)^(k)(0)=0, k=0,1,2, …,n.

  
  As (V,ø) is an n-correspondence, the equation ℓ∘(t)=0 has degree ≤ n therefore
    ℓ∘ vanishes identically. As the vector v is arbitrary, we deduce that ℓ vanishes on (V,ω). 
  This inclusion being true for any hyperplane containing V,ω_n,
  we  get that V,ø⊂ V,ω_n.
  This proves the lemma.
  
 
  


 §.§ The spans stabilise

We return to the situation of  subsection <ref> and consider a decreasing chains of subsets of ^N=^d ×^e:

    X_0 ⊃ X_1 ⊃ X_2 ⊃…⊃ X_∞ :=∩_n X_n .

with
X_n^∘=X_n  
 and consider a formal power series
 
    f(x,y) =∑_ I ∈^d a_I(y)x^I ∈{ y} [[x]], f(0)=0

 and define
 
    f_n(x,y) =∑_|I| ≤ n, I ∈^d a_I(y)x^I ∈{y}[x]

We denote their
zero-sets by

    V_n:={(x,y) ∈ X_n |  f_n(x,y)=0}⊂ X_n


We also have to consider

    V_∞:={(x,y) ∈ X_∞ |  f(x,y)=0}⊂ X_∞







One has  
    V_n,0⊂ V_n+1,0 .


 We have
  
    V_n,0 = V_n,0_n= V_n+1,0_n⊂ V_n+1,0_n+1= V_n+1,0.

 For the first and last equalities we used that V_n is an n-correspondence
  (for n and n+1), the second equality is due to the fact that
  
    f_n+1(x,y)=f_n(x,y)+O(x^n+1)
 The middle inclusion is
  by the definition of jet-span.


As a consequence, the sequence V_n,0 stabilises, say at level ρ
and we define

    W_∞ := V_n,0, n ≥ρ .



For n ≥ρ, we have:

    V_∞,0⊂ V_n,0= W_∞


  

    Take (x_0,y_0) ∉ W_∞. By definition of W_∞, this means that for any n ≥ρ,
    the point (x_0,y_0) does not belong to V_n,0 and therefore is not in V_n, so we have
    f_n(x_0,y_0) ≠ 0 for n ≥ρ. Consequently, for n ≥ρ, the holomorphic map-germs
   
    ϕ_n:(^d,x_0) →^e, x ↦ f_n(x,y_0)=∑_|I| ≤ n a_I(y_0)x^I

are  non-zero at x=x_0.

 We now use the classical Hurwitz lemma (see e.g. <cit.>): Consider a sequence of holomorphic functions ϕ_n converging in the C^0-topology and assume that the limit ϕ_∞ is not the zero-function. Then:
  if the functions ϕ_n are non-zero at a point, then the limit ϕ_∞ is also non-zero at this point
  [This Hurwitz lemma, sometimes called the Hurwitz theorem, is a straightforward consequence of the Cauchy integral formula.].

As f_n(x_0,y_0) ≠ 0, the coefficients a_I are not all equal to zero and therefore the holomorphic map-germ

    ϕ_∞:(^d,x_0) →^e, x ↦∑_I ∈^d a_I(y_0)x^I

is not identically zero. From the Hurwitz lemma we conclude that  (x_0,y_0) ∉   V_∞ and therefore  
    (V_∞,z_0) ⊂ W_∞, z_0 =(x_0,y_0)
 


 

 §.§ Proof of Theorem <ref>

 We keep the same notations and let 
 
    g:(× W_∞,0) → (^e,0)

 be the restriction of f to W_∞. By Lemma <ref>,
 we have 
 
    (V(f),0)=(V(g),0).
 We choose a Whitney extension G of g. By the implicit function theorem the zero locus of G is the graph of a function-germ 
 
    :̱(^d,0) → (W_∞,0), x ↦(̱x)

 and we may extend the image to ^e to get a Birkhoff function

    B:(^d,0) → (^e,0), x ↦(̱x)

  The Taylor expansion of B at the origin is the Birkhoff series whose partial derivatives generate W_∞
  by definition. Hence the function B is KMS non-degenerate. This concludes the proof of the theorem.



§ APPLICATION TO THE HERMAN CONJECTURE

We give a brief outlook of our proof of the Herman invariant tori conjecture, details will be published elsewhere, the reader may consult <cit.> for further details.


 §.§ The frequency map

In KAM theory, the invariant tori of a Hamiltonian system are parametrised by the frequencies of motions which are not very well approximated by the standard integral lattice ^e.
In order to measure how a frequency vector y ∈^e is approximated by the lattice, one considers the collection of resonant hyperplanes (_) with 

    _={ y ∈^e: (y,) =0 }, ∈^e

where 

    (y,)=∑_j=1^e y_j _j

is the complexification of the Euclidean scalar product. We then fix a real positive sequence
(_), ∈^e, which converges to zero not too fast
and consider tubular neighbourhoods of hyperplanes

    T_={ y ∈^e: |(y,)| < _}, ∈^e

 We now define the sequence (X_n) by:

    X_n=^e ∖⋃_≤ n T_

and this set parametrises the invariant tori, in the sense that there is a Birkhoff function

    B: U →^e

called the frequency map whose fibres above points of X_∞=⋂_n X_n are (complexifications) of invariant tori. In <cit.>, we constructed B as a map obtained from a
correspondence as we now explain.



 §.§ The small denominators ring

We consider the polynomial ring [ω]=[ω_1,ω_2,…,ω_d] and construct a ring SD_α of small denominators at α∈^d. The ring SD_α is a subring of the field (ω)=(ω_1,ω_2,…,ω_d) of rational functions 
 defined by 
localisation of [ω]=[ω_1,ω_2,…,ω_d] with respect with the multiplicative subset S generated by all linear polynomials (+ω, J), J ∈^n∖{0}:

    SD_α:=[ω]_S:=[ω,1/(+ω,J), J ∈^n∖{0}]
    ⊂(ω) .

The elements of this ring may have poles along the resonance hyperplanes

    H_J:={ω∈^d: (+ω,J)=0 }

We will construct a normal form inside the algebra

    R:=SD_α[[τ,p,q]] ⊂[[ω,τ, p,q]],

a subalgebra of the power series in 4d variables 

    ω_1,ω_2,…,ω_d,τ_1,…,τ_d,q_1,…,q_d,p_1,…,p_d .

We provide R with the standard Poisson bracket, so that the Poisson centre of
R is the ring

    R_0:=SD_[[τ]] ⊂[[ω_1,…,ω_d,τ_1,…,τ_d]].

We endow this ring of quasi-homogenous filtration with weights

    w(q_i)=w(p_i)=1, w(τ_i)=2, w(ω_i) =0

We use the notations [f]_i^j for the sum of terms of weighted degree ≥ i and <j and omit the indices if i=0 or j=+∞.
We also use the notation O(n) for terms of degree ≥ n.



 §.§ The Hamiltonian normal form iteration

Starting from a Hamiltonian 

    H=∑_i=0^d α_i p_iq_i+O(3),

we first form the ω-extension of H:

    F_0=∑_i=0^d (α_i+ω_i) p_iq_i+O(3),


Then we solve a homological equation of the form:

    v_0(A_0)=[F_0]_3^4 +t_0,    t_0 ∈  R_0+ I^2 ,
 
where I is the ideal generated by the  p_iq_i-τ_i and v_0 is a Poisson derivation.
The definition might seem involved but it produces a sequence of biholomorphic maps e^-v_i which are defined globally and not "blocked by resonances."

Exponentiating the Poisson derivation v_0, we produce a Poisson automorphism  e^-v_0. The application of e^-v_0 transforms F_0 into F_1, where the cubic term is  removed modulo an element of R_0+I^2 ; we put A_1=A_0+t_0. For the n=0 case, it turns out that t=0 but at the next level we have to solve

    v_1(A_1)=[F_1]_4^6 + t_1 ,    t_1 ∈ R_0+I^2
 
for the degree 4 and 5 part of F_1 on A_1 and, as a general rule t_1 ≠ 0. 

Then the application of e^-v_1 to F_1 produces F_2, where now these terms of degree 4 and 5 are removed, but certain terms in R_0+I^2 are introduced. These remaining terms we add to A_1 and obtain A_2. Next we solve the homological equation for the terms of degree 6,7,8,9 of F_2, but now on  A_2, etc. Thus we obtain a by iteration a sequence of triples

    (F_n, A_n, v_n ),    n=0,1,2,…

We have correspondences defined by the rational functions 

    [e^-v_n… e^-v_0ω_i]^2^n= ∑_, || ≤ 2^n a_(ω) τ^, a_∈ SD_

 
 precisely of the kind envisioned by Poincaré!  This power series is X-convergent, the partial sums define complex variety and the limit variety
 parametrises the pairs consisting of invariant tori and associated frequencies. 
 
 The theorem of this paper shows that the limiting correspondence contains the graph of a KMS
 non-degenerate map.  Using an adequate variant of the Kleinbock-Margulis theorem (<cit.>), one deduces the fact that the invariant tori form a set of positive measure.  The convergence
 of the iteration scheme proposed  is subtle, but it turns out that it can be
 streamlined using the theory of Banach functors developed in <cit.>.  In that
 set-up, the non-degeneracy argument is easy and, in fact, as this paper shows, “almost trivial”. 

 
amsplain



 Paper Title: A method for verifying the generalized Riemann hypothesis.





G. Hiary, S. Ireland, M. Kyi]Ghaith Hiary, Summer Ireland, Megan Kyi

    GH: Department of Mathematics, The Ohio State University, 231 West 18th
    Ave, Columbus, OH 43210, USA

hiary.1@osu.edu

    SI: Department of Mathematics, The Ohio State University, 231 West 18th
    Ave, Columbus, OH 43210, USA

ireland.118@buckeyemail.osu.edu

    MK: Oberlin College, 135 West Lorain Street, Oberlin, OH 44074-1081, USA

mkyi@oberlin.edu

[2020]Primary: 11M06, 11Y35.


§ ABSTRACT 
    Riemann numerically approximated at least three zeta zeros. According to Edwards, Riemann even took steps to verify that the lowest zero he computed was indeed the first zeta zero. This approach to verification is developed, improved, and generalized to a large class of L-functions. Results of numerical calculations demonstrating the efficacy of the method are presented.


A method for verifying the generalized Riemann hypothesis
    [
    
=========================================================



§ INTRODUCTION


Let s=σ+it be a complex variable, where σ and t are real numbers. 
The Riemann zeta function ζ(s) is defined by the Dirichlet series

    ζ(s)=∑_n≥ 11/n^s,

which converges absolutely in the half-plane σ >1.
Zeta can be analytically continued to the entire complex plane except for a simple pole at s=1, and has zeros (i.e., roots) at s=-2,-4,-6,…, which are called the trivial zeros. Zeta also has an infinite number of nontrivial zeros ρ=β+iγ in the critical strip 0< σ< 1, none of which is real. We call |γ| the height of ρ and order the ρ's by increasing height.
The trivial and nontrivial zeros account for all the zeta zeros.
The Riemann Hypothesis (RH) is that all the ρ's are on the critical line σ=1/2, 
or equivalently that β=1/2 for all ρ. 

It is frequently asserted that Riemann numerically approximated the first few zeta zeros by hand, citing unpublished notes by Riemann. See in particular Edwards <cit.> as well as the Clay Mathematics Institute page <cit.>.
As Figure <ref> shows, 
Riemann numerically approximated three zeta zeros on the critical line, 
corresponding to those with ordinates (i.e., imaginary parts)

    γ_1= 14.1347251417…,
    γ_2=21.0220396387…,
    γ_3=25.0108575801….

Riemann approximated γ/(2π) rather than γ as the former 
quantity appeared naturally in various formulas.

The closest approximation of γ_1/(2π) that we found in Riemann's notes was 2.250466, so that γ_1 is ≈ 14.140095.[This differs from what is stated in <cit.> which gave the approximation 14.1386.] For γ_2 and γ_3, Riemann computed the approximations 3.287195 and 
4.0287, respectively. Both of these approximations were 
noticeably far from the true values 3.34576152… and 3.98060161…, and 
as can be seen in Figure <ref>, Riemann had other intermediate approximations that were 
slightly better. 
Nevertheless, Riemann's approximation of γ_1, which was
long unknown to the outside world, 
remained closest to the true value of γ_1 for nearly five decades.[As far as we can tell, the first 
circulated approximation of γ_1 was in 1887 by 
Stieltjes <cit.> who gave the approximation 14.5.
Eight years later, Gram <cit.>
gave the approximation 14.135, which Gram <cit.> improved to 14.13472 
in 1903. Around the same time, Lindelöf <cit.> devised a different method 
to approximate the ρ's and
proved that 14≤γ_1≤ 14.25.]

According to Edwards <cit.>, Riemann even attempted 
to verify that Riemann's numerical approximation of 1/2+iγ_1 indeed
corresponded to the first zeta zero (i.e., to the ρ with smallest positive ordinate). 
This verification relied on the Hadamard product for ζ(s) together with
a positivity argument and  a known special value of zeta. 
However, unlike Riemann's method to numerically compute pointwise values of ζ(s), which 
became a standard method known as the Riemann–Siegel formula <cit.>,
Riemann's approach to  verifying the RH remained little known. 
It is worth remarking, though, 
that Gram <cit.> 
considered the power series of the logarithm of the Hadamard product, like Riemann did, 
but often appeared to assume the RH. One may also 
compare the Riemann approach to verification with the Li 
criterion <cit.> for the RH equivalence. 

After Riemann's 1859 paper, various efficient 
methods for verifying the RH were derived. 
Backlund <cit.> 
devised a verification method that relied on
    a clever application of 
    the argument principle from complex analysis 
 together with the Euler–Maclaurin summation for ζ(s).
    This was eventually surpassed by 
    a highly efficient method due to Turing <cit.>, 
   which has since become the standard method for verifying the RH, 
     provided one is high enough on the critical line.
     


In comparison, the Riemann method, cited in Edwards <cit.>, 
is time consuming at large heights. It can be expected to require
∼1/2π^2(t_0log t_0)^2 initial zeta zeros to verify the RH up to height t_0.
  Nevertheless, this method is reasonably efficient at low heights[For example, 10 zeta zeros suffice to verify the RH up to height γ_1 via this method, 
and 51 zeta zeros suffice to verify the RH up to height γ_2.] while 
  offering great simplicity.
    Therefore, it is worthwhile to generalize the Riemann method 
    to families of L-functions
    where even the “first” zero of L(s) is deeply interesting. Such a generalization  is one of the main goals of this paper. 
    
    Specifically, rather 
    than fall back on a generalized Backlund method, which would require using 
    a numerically-involved application of the argument
    principle, 
    we re-examine and develop the Riemann method in a more general setting. 
    Our generalization
    works naturally with 
    already available databases and software for L-functions. 
    Our generalization
     is also simple to derive and justify, 
    requiring a single
    numerical evaluation of a logarithmic derivative of the L-function at a special point in the region of absolute convergence.

    Although the main goal of this paper is to provide a simple RH verification for a large class of L-functions at low heights, a secondary goal is to improve the Riemann method for zeta 
    so that it functions efficiently at large heights. This results in a conceptually straightforward verification method that can verify the RH over larger windows.  Specifically, given zeros data in a window of size τ around height y, the improved method in Theorem <ref> 
 is expected to succeed in verifying the RH in a window of size η≫τ/√(log y) around y.

To illustrate out main results, let us state two
corollaries. 
Corollary <ref> provides an example of
an RH verification test for low zeros of 
 the Dirichlet L-function L(s,χ_d), where χ_d is any real primitive character of fundamental discriminant d. 
This corollary is obtained 
from Theorem <ref>, part (i), on setting δ=-1 and m=1, 
and using the formula for w_1,δ in Corollary <ref>. 

Note
the required value of the logarithmic derivative of L(s,χ_d) 
that appears in Corollary <ref>
is well inside the region of absolute convergence of L(s,χ_d). 
So, the required value can be computed easily by truncating the 
Dirichlet series for the logarithmic derivative, even if d is very large.
Lemma <ref> furnishes an explicit bound on the corresponding truncation error. 

To state the next two corollaries, we will make use of the following the quantity.

    ι(η):=min(1/1+η^2+2/4+η^2, 12/9+4η^2).


Let d be a positive fundamental discriminant, τ be a real positive number, 
and 𝒵 be a set of nonempty disjoint subintervals of the form [γ_-,γ_+]⊆ [0,τ] or of the form [-γ_0,γ_0]⊆ [-τ,τ]. Suppose that L(1/2+it,χ_d) has a zero of odd multiplicity in each subinterval in 𝒵.
Further, define  

    C(𝒵):=
    ∑_[γ_-,γ_+]∈𝒵12/9+4γ_+^2
    + ∑_[-γ_0,γ_0]∈𝒵6/9+4γ_0^2.

Let λ_0=0.57721566… be the Euler constant. For any real positive number η, if

    2ι(η) +C(𝒵)
     > 1/2logd/π e^λ_0+ L'/L(2,χ_d),

then then RH holds for all the nontrivial zeros of L(s,χ_d) with positive height ≤η.


    Here, one may think of τ as the width of the window where known zeros data is available, 
    and of η as the width of the window where one would like to verify the RH. The quantity C(𝒵) is the minimal contribution from known zeros (i.e. from supplied zeros data), 
    and ι(η) is the minimal contribution of a hypothetical counter-example of positive height ≤η. The displayed inequality indicates that a contradiction has been reached, so that a hypothetical counter-example of positive height ≤η cannot exist. 

Similarly, by combining Theorem <ref>, part (i) and Corollary <ref>, 
we obtain an RH verification test for zeta zeros at large heights. 
This is stated in Corollary <ref>.
But in this case we can improve the basic verification test substantially 
  by considering the behavior 
of S(u), which is the fluctuating part of the counting function of zeta zeros - see (<ref>).
Specifically, in Theorem <ref>, we incorporate the explicit bounds 

    |S(u)|≤ℓ(u)    and    |∫_u_0^uS(ν) dν|≤ℓ_1(u),

where, according to <cit.>, we may take 

    ℓ(u)    := 0.112 log u + 0.278 loglog u + 2.510, 
    ℓ_1(u)    := 0.059 log u+2.067,

provided u is large enough. (u>u_0>168π suffices.)
We note that an explicit bound on |∫_u_0^uS(ν) dν| is a main ingredient in the Turing method as well, but we use this bound differently in our case. 
Also, without additional knowledge or analysis, the explicit bound on |S(u)| is typically 
far more impactful for us than the explicit bound on |∫_u_0^uS(ν) dν|.
 
We will make use of the following notation and quantity. Let
g(s)=(s-1)ζ(s), ψ_0 denote the digamma function, and define

    κ(y,τ):= 0.57/τ y^2+3log 2/2π y
    +2log(y/2π)/πτ^3+3log(y/2π)/π y
    +12 ℓ(2y)/y^2 + 6 ℓ_1(2y)/τ^3.
 


Let y and τ be real numbers such that 3 <τ≤ y/2 and 336π < y-τ. 
Let 𝒵 be a set of nonempty disjoint subintervals of the form [γ_-,γ_+]⊆ [y-τ,y+τ]
such that y does not belong to any of the subintervals in 𝒵.
Suppose that ζ(1/2+it) has a zero of odd multiplicity in each subinterval in 𝒵.
Further, define 

    D(𝒵):=∑_[γ_-,γ_+]∈𝒵
    γ_->y6/9+4(γ_+-y)^2
    +∑_[γ_-,γ_+]∈𝒵
    γ_+< y6/9+4(y-γ_-)^2.

For any real positive number η≤ y, if
 
    ι(η) + D(𝒵) + 3/2πτlogy/2π - 6 ℓ(2y)/2τ^2    - κ(y,τ) >   
         - 1/2logπ +1/2 ψ_0(2-iy/2)+
        g'/g(2-iy),

then RH holds for all the nontrivial zeros of ζ(s) with height in [y-η,y+η]. 



Here, one may think of y as very large, τ is large but much smaller than y, 
and with η somewhat smaller than τ. The expression for κ(y,τ) 
is obtained from Theorem <ref> by setting x=-1 and c=y/2.


Like before,
the special value of the logarithmic derivative of g(s) appearing in Corollary <ref>
is well inside the region of absolute convergence of ζ(s).
So this value can be approximated easily and fairly accurately 
via a truncated sum over primes and prime powers
using our Lemma <ref>, even at very large heights.

Our main theorems, Theorem <ref> and Theorem <ref>, additionally
 enable verifying the simplicity of zeros in a given range as well as 
 verifying the completeness of a given list of zeros.
In addition, Theorem <ref> gives 
a counterpart that allows
one to still draw a conclusion 
in some situations where the RH might not be verified using the
Turing method. For example, if the given zeros list is 
incomplete (i.e. there is a zero with ordinate in [y-τ,y+τ] that is missing from the list), then the 
the Turing method 
might not prove that the zeros list is indeed incomplete. 
In this case, the counterpart in Theorem <ref> 
will typically enable proving that the given zeros list is indeed incomplete.

Lastly, it completely reasonable to expect that a similar method to the one described here may be derived using the framework of the explicit formula <cit.>. By choosing a suitable test function in the explicit formula, one may even accelerate the convergence of the associated series over the prime and prime powers. 
At the same time though one must ensure, under no assumption, that the individual terms in 
the sum over the zeros appearing in the explicit formula are nonnegative. We favored 
the current derivation due to its simplicity, its historical connection, and because we already have good control 
over the convergence of the said series in the region of absolute convergence. 
Additionally, the current derivation gives us access to several useful exact values and exacting relations as well 
as to long-studied sums in the theory of the Riemann zeta function, which benefits the practicality of our derivation. 
For example, we can directly benefit from exact values of the polygamma function and, if we wish, 
of exact values of L-functions at special points such as the class number formula for Dirichlet L-functions. 




Overview.
In <ref>, we provide background and set up some notation.
In <ref>, we outline the Riemann approach to verifying the RH following the description in 
<cit.>. 
In <ref>, we generalize the Riemann approach to a class of L-functions with real Dirichlet coefficients.
In <ref>, we treat the case of ζ separately,  
both because ζ(s) is outside our class of L-functions (in view of the pole at s=1) and 
because our focus for zeta will be on large heights.
In <ref>, we discuss substantial improvements in the case of zeta.
In <ref>, we present results of numerical computations 
implemented in interval arithmetic
for a variety of examples of L-functions.



§ BACKGROUND AND NOTATION


Using the Dirichlet series (<ref>), we see that

    ζ(s)=ζ(s).

So, ρ=β+iγ is a zeta zero if and only if the complex conjugate ρ=β-iγ is a zeta zero, or equivalently the ρ's are symmetric about the real axis. 
The ρ's are also symmetric about the critical line. This 
is seen by using the zeta functional equation, which 
in its simplest form states that
the entire function 

    ξ(s):=π^-s/2Γ(s/2+1)(s-1)ζ(s),

where Γ is the Gamma function[The poles of Γ(s/2+1) are all simple 
and coincide with the trivial zeros of zeta, all of which are simple as well. So, the poles of Γ(s/2+1) cancel the trivial zeros of zeta. 
The simple pole of zeta at s=1 coincides with the zero of the factor s-1 in the definition of ξ.], satisfies the functional equation 

    ξ(s)=ξ(1-s).

Therefore, ξ(s) is even about s=1/2. 
For example, ξ(0)=ξ(1) = -ζ(0)=1/2. 
Since Γ and π^-s/2 have no zeros at all,
the zeros of ξ are the same as the nontrivial zeros of ζ. 
Hence, by the functional equation (<ref>), ρ is a zeta zero if and only if 1-ρ is a zeta zero. 

Furthermore, by the functional equation (<ref>) and the symmetry relation (<ref>), 

    ξ(1/2+it)=ξ(1/2-it)=ξ(1/2+it).

So, ξ is real-valued on the critical line (as well as on the real axis).
It follows by the intermediate value theorem that the simple (or odd multiplicity) 
nontrivial zeta zeros on the critical line correspond to sign changes of ξ(1/2+it). In particular, one can numerically prove the existence of zeta zeros of odd multiplicity
on the critical line by detecting sign changes of ξ(1/2+it).



§ RIEMANN AND VERIFYING THE RH

Being an entire function of order 1, ξ has 
a Hadamard product given by

    ξ(s)=ξ(0)∏_ρ(1-s/ρ),

where 
the product is taken by pairing the terms for ρ and ρ (or pairing the terms for ρ and 1-ρ), which ensures correct convergence. 
Starting with  (<ref>), Riemann obtained the following formula

    ∑_ρ1/ρ=v_1    where     
    v_1:=1/2λ_0 +1-1/2log 4π,

and the sum over the ρ's is executed by pairing the terms for ρ and ρ. 
Therefore,

    v_1=2∑_γ>0 1/ρ.

As seen in Figure <ref>, Riemann correctly computed the value of v_1 
up to 20 digits,
obtaining v_1=0.02309570896612103381….


According to Edwards <cit.>, Riemann even 
attempted to use the 
numerical value of v_1 to verify that the Riemann approximation of ρ_1=1/2+iγ_1 
indeed corresponded to the first zeta zero (zeta zero of lowest height). This attempt is described essentially as follows.

Using the first 10 zeros in the upper half-plane, 2  (ρ_1^-1+…+ρ_10^-1) ≈ 0.0136.
On the other hand, 
if there is a zero ρ_0 in the upper half-plane of height <γ_1, 
then there must be a second such zero. This is because either 
ρ_0 is off the critical line, in which case 1-ρ_0 is a distinct zeta zero in the upper half-plane that is also of 
height <γ_1. Or ρ_0 is on the critical line, in which case, 
considering that ξ(1/2+it)  has the same sign at both t=0 and t=14.1,
there must be a second zero on the critical line with a positive ordinate <γ_1.[More precisely, 
the argument in <cit.> 
only works if ρ_0 has height < 14.1. 
Since the possibility that ρ_0 has height ≥ 14.1 is not yet ruled out, 
this argument  
does not force the existence of a second zero on the critical line in this case.] 
Therefore, if ρ_0 existed, then it would force an additional contribution of at least 2 (ρ_1^-1),
 causing the zeros sum to exceed v_1 and hence gives a contradiction.




Although not stated explicitly, it is critical to the last part of the argument that the terms 

    1/ρ= β/β^2+γ^2
 
are all nonnegative. 
This ensures that the tail of the zeros sum contributes a nonnegative amount to v_1. 
Therefore, we can drop the tail of the zeros sum and still obtain 
a valid lower bound on v_1. 

More generally, in this paper, 
we will consider the behavior of the function 

    ϕ(β,η,x):= β-x/(β-x)^2+η^2+ 1-β-x/(1-β-x)^2+η^2.
 
If z=x+iy is a complex number then we have

    [1/ρ-z+1/1-ρ-z]= 
    ϕ(β,γ-y,x).

Note that ϕ(β,η,x) is nonnegative for 0≤β≤ 1 and x≤ 0. 
To analyze the behavior of ϕ in detail, we will often invoke the following lemma.


    Let β be a real number such that 0≤β≤ 1. Let 
    x be a real nonpositive number, and let η be a real positive number. Then ϕ(β,η,x)≥ 0. Furthermore, we have the following.
    

      
      
        
  (i) If η≤√(x(x-1)/3), then ϕ(β,η,x) is minimized at β=1/2.
        
  (ii) If η > √(x(x-1)/3), then ϕ(β,η,x) is minimized at β=0 (or β=1).
        
  (iii) If η> 1-2x/2√(3), then ϕ(β,η,x) is maximized at β=1/2.
        
  (iv) ∂/∂ uϕ(β,u,x) is negative. Additionally, if u > 1-2x/2√(3) then ∂/∂ uϕ(1/2,u,x) is increasing, and if u>2-2x/2√(3) then ∂/∂ uϕ(0,u,x) is increasing.
    
    



    See <ref>.




§ GENERALIZATION TO A CLASS OF L-FUNCTIONS

 
In the sequel, 
we use the analytic normalization of L-functions, so the critical line is σ=1/2.
We consider L-functions of order 1 only.
The following notation and assumptions are used throughout this section.
Let L(s) be a Dirichlet series 

    L(s)=∑_n≥ 1a(n)/n^s,

absolutely convergent in the half-plane σ>1.
We suppose that the Dirichlet coefficients a(n) are real, 
so that 

    L(s) = L(s),
 
and the zeros of L(s) 
must be symmetric about the real axis. 

Following the notation in Booker <cit.>,
specialized to our context[In particular, we require that the μ_j are real and 
μ_j≥ 0 instead of (μ_j)≥ -1/2. We also write the formulas 
for Γ_ℝ(s) and f(z) explicitly as 
π^-s/2Γ(s/2) and f(z), respectively, as well as
drop a scaling factor by N^-1/4 in the definition of γ(s) in <cit.> as this
does not interfere with any of our calculations.], we state a number of assumptions satisfied 
by the set of L-functions we consider.
L(s) has an Euler product of degree r absolutely convergent in the half-plane σ > 1,

    L(s)=∏_p prime1/(1-α_p,1p^-s)⋯ (1-α_p,rp^-s),

where the α_p,j satisfy the conditions in <cit.>.  We will further assume
that |α_p,j|≤ 1.
Note that by the absolute convergence of the Euler product, L(s) has no zeros in the half-plane σ>1.

Suppose further there 
are positive integers r and N, a complex number ϵ of modulus 1, and real nonnegative numbers μ_1,…,μ_r, such that the function ξ_L(s) defined by

    ξ_L(s) := γ(s)L(s),    γ(s) :=ϵ N^s/2π^-sr/2∏_j=1^r Γ(s/2+μ_j/2),

extends to an entire function and satisfies the functional equation

    ξ_L(s)= ξ_L(1-s).


Note that by the functional equation, ξ_L(1/2+it) is real.
Also, ξ_L is real on the real axis.
If ϵ=± 1, 
then the functional equation simplifies to ξ_L(s)=ξ_L(1-s) which means that 
ξ_L(1/2+it) is even in t. While if ϵ=± i, then
ξ_L(s)=-ξ_L(1-s) which means that ξ_L(1/2+it) is odd in t, 
and hence must have a zero of 
odd multiplicity at t=0.


Since ξ_L(s) is entire, L(s) must have zeros at the poles of γ(s), 
which are the trivial zeros of L(s). 
Since L(s) has no zeros in the half-plane σ>1,  
it follows by the functional equation that the trivial zeros of L(s) in
σ<0 have the same multiplicities as the poles of γ(s). Moreover, the
 nontrivial zeros of L(s), which we 
 denote by ρ=β+iγ, are in the critical strip 0≤σ≤ 1.

We assume L(1) 0, so that ξ_L(1) 0, and hence ξ_L(0) 0.
Therefore, the zeros of ξ_L(s) are exactly the nontrivial zeros of L(s). 
Also, just like ξ(s), ξ_L(s) being of order 1 has a Hadamard product

    ξ_L(s)=ξ_L(0)∏_ρ(1-s/ρ),

where we pair the terms for ρ and ρ (or for ρ and 1-ρ). 
 The RH for L(s) is the assertion that all the ρ's are on the critical line σ=1/2.

To state the next proposition, we recall the j-th order polygamma function ψ_j(s), 
defined as the j-th derivative of 
ψ_0(s) = Γ'(s)/Γ(s). 
Also, for any real number δ<1  such that L(1-δ) 0, let us write 

    log L(s-δ)=∑_j≥ 0 d_j,δ (s-1)^j,     d_j,δ=1/j![d^j/ds^jlog L(s-δ)]_s=1

for s sufficiently close to 1.



    Let k be a positive integer. 
    Let δ be a real number such that δ<1
    and ξ_L(δ) 0. 
    Define
    
    w_k,δ:=∑_ρ1/(ρ-δ)^k,

    where the sum is ordered by pairing each term with its conjugate.
    Then w_k,δ is a real number.
    If k>1, then
    
    w_k,δ= (-1)^k-1[1/2^k (k-1)!∑_j=1^rψ_k-1(1/2-δ/2+μ_j/2)+k d_k,δ].

    And if k=1, then the same formula holds but there is an additional term of 
    
    1/2log N- r/2logπ.



Since δ is a real number and the ρ's are symmetric about the real axis, w_k,δ is real.
By the Hadamard product for ξ_L,

    logξ_L(s+δ) = logξ_L(δ)-∑_k≥ 1w_k,δ/k s^k,

provided s is sufficiently close to 0. Therefore,

    -w_k,δ/k=1/k![d^k/ds^klogξ_L(s+δ)]_s=0.

By the functional equation (<ref>), 

    1/k![d^k/ds^klogξ_L(s+δ)]_s=0=(-1)^k/k![d^k/ds^klogξ_L(s-δ)]_s=1.

On the other hand, recalling the definition of ξ_L(s),

    logξ_L(s) = logϵ+s/2log N -sr/2logπ+∑_j=1^r logΓ(s/2+μ_j/2)+log L(s)

for s away from zeros or poles of both sides. Therefore, replacing s with s-δ, 
and using the series expansion (<ref>), we obtain

    1/k![d^k/ds^klogξ_L(s-δ)]_s=1=    1_k=1(1/2log N-r/2logπ)+ 
       1/2^k k!∑_j=1^rψ_k-1(1/2-δ/2+μ_j/2) + d_k,δ,

where 1_k=1 is the indicator function of the condition k=1.
Substituting (<ref>) back into (<ref>),
then back into (<ref>), yields the proposition.


Since our numerical experiments in <ref> 
will focus on the case k=1, we provide a version of Lemma <ref> in this special case.
 
When k=1, we have
    
    w_1,δ=1/2log N- r/2logπ+
        1/2∑_j=1^rψ_0(1-δ+μ_j/2)+
         L'(1-δ)/L(1-δ).



Let us note that many special values ψ_0(s) can be expressed exactly 
in terms of known constants.[For example, when s=1/2, 
    ψ_0(1/2)=-2log 2-λ_0, ψ_1(1/2)= -π^2/2,
    ψ_2(1/2) =-14ζ(3), ψ_3(1/2) = π^2,
and more generally for j≥ 1, 
ψ_j(1/2)= (-1)^j+1j!(2^j+1-1)ζ(j+1). As another example, when s=1, we have ψ_0(1)=-λ_0.] 
In general, there are
efficient ways for computing ψ_0(x) for x>0; 
see for example <cit.> for a discussion of methods to compute Γ, ψ_0, and related functions.
Therefore, for the purpose of computing w_1,δ, we may focus our attention on the logarithmic derivative 
of L(s) at s=1-δ. 
The next lemma supplies a simple formula for doing this, provided δ<0. 
We make use of the following notation: 
if n=p^m for a prime p and a natural number m, then

    Λ_L(n):= log p ∑_j=1^r α_j,p^m,

and we set Λ_L(n)=0 otherwise. In particular, 
since |α_j,p|≤ 1, 

    |Λ_L(n)|≤ r Λ(n),
 
where Λ(n) is the von Mangoldt function. This is defined by 
Λ(n) = log p if n=p^m for a prime p and a natural number m, 
and Λ(n)=0 otherwise.


Let K≥ 18 be an integer. If δ<0, then

    L'(1-δ)/L(1-δ) = -∑_k= 1^K Λ_L(k)/k^1-δ + ℛ_L(K,δ),

where

    |ℛ_L(K,δ)|< r K^δ/δ(2.85·2δ-1/log K-1).



    Suppose σ >1. By the Euler product for L(s),

    L'/L(s) = -∑_k≥ 1Λ_L(k)/k^s.

So, by Stietljes integration and the bound |Λ_L(k)|≤ r Λ(k), 
the tail ℛ_L(K,δ) 
of the Dirichlet series (<ref>) for k>K and s=1-δ satisfies

    |ℛ_L(K,δ)|≤ 
         r∫_K^∞1/u^1-δ dψ(u)   where  ψ(u) = ∑_k≤ uΛ(k).

Using integration  by parts, 

    ∫_K^∞1/u^1-δ dψ(u) = -ψ(K)/K^1-δ+(1-δ)∫_K^∞ψ(u)/u^2-δ du.

Furthermore, by <cit.>, we have for u≥ K 
the double inequality 

    0<u(1-2.85/log K) ≤ψ(u) ≤ u(1+2.85/log K).
 
Substituting this into 
(<ref>), then back into (<ref>), and integrating yields the result.



A simpler version of Lemma <ref> is obtained by using the trivial bound
|Λ_L(k)|≤ rlog k. This gives 

    |ℛ_L(K,δ)|≤ 
        r∑_n>Klog k/k^1-δ< r∫_K^∞log u/u^1-δ du= 
        r K^δ·1-δlog K/δ^2.

Although usually not as precise as Lemma <ref>, 
this estimate is sharper than Lemma <ref> 
if δ is very large compared to log K.



Lemma <ref> generalizes easily to higher order 
logarithmic derivatives of L(s) at s=1-δ. For example,

    [d^2/ds^slog L(s)]_s=1-δ = 
    -∑_k= 1^K Λ_L(k)log k/k^1-δ + ℛ_L,2(K,δ),

where

    |ℛ_L,2(K,δ)|< r K^δ(2.85-log K 
    +1/(1-δ)-δlog K/δ^2(1-δ)(1+2.85/log K)
    ).



Theorem <ref> next is our main result in this section. Unlike the case of zeta, where none of the ρ's is real, L(s) might have real nontrivial zeros. So,
care is needed to allow for this possibility. 
The following lemma will facilitate the proof of Theorem <ref>. Recall 
that the function ϕ was defined in (<ref>), and that

    [1/ρ-δ+1/1-ρ-δ]= 
    ϕ(β,γ,δ).




    Let δ be a real nonpositive number and let τ be a real positive number. Let 𝒵 be a set of nonempty disjoint subintervals of the form [γ_-,γ_+]⊆ [0,τ] or of the form [-γ_0,γ_0]⊆ [-τ,τ]. Suppose that ξ_L(1/2+it) has a sign change in each subinterval in 𝒵.[This means ξ_L(1/2+iτ_1)<0<ξ_L(1/2+iτ_2) for some τ_1,τ_2 in each subinterval in question.]. Define 
    
    C(𝒵,δ):=∑_[γ_-,γ_+]∈𝒵1-2δ/(1/2-δ)^2+γ_+^2 
        + ∑_[-γ_0,γ_0]∈𝒵1/2-δ/(1/2-δ)^2+γ_0^2.

    For any real positive number η, any positive integer m, and with ϕ as in (<ref>),
    define
    
    f_1(η,δ,m)    :=2m·min(ϕ(0,η,δ),ϕ(1/2,η,δ)), 
    
           f_2(η,δ,m)    := m·ϕ(1/2,η,δ), 
    
           h_1(δ,m)    := m·ϕ(1/2,0,δ), 
    
           h_2(δ,m)    :=m/2·ϕ(1/2,0,δ), 
    
           F(η,δ)    := min(2·ϕ(0,η,δ),ϕ(1/2,η,δ),1/2·ϕ(1/2,0,δ)).

    Then, we have the following, where zeros are counted with multiplicity in all cases.
    
        
  (i) If
    f_1(η,δ,m) +C(𝒵,δ) > w_1,δ,
    then there are strictly fewer than 4m non-real ρ's off the critical line of height ≤η.
     
  (ii) If
    f_2(η,δ,m) +C(𝒵,δ) > w_1,δ,
    then there are strictly fewer than 2m non-real ρ's on the critical line of height ≤η not accounted for in 𝒵.
      
  (iii) If h_1(δ,m) +C(𝒵,δ) > w_1,δ,
    then there are strictly fewer than 2m real ρ's off the critical line.
     
  (iv)If
    h_2(δ, m) +C(𝒵,δ) > w_1,δ,
    then a zero at the central point s=1/2 has multiplicity strictly less than m.
    
  (v) If F(η,δ) +C(𝒵,δ) > w_1,δ,
    then the list of zeros in 𝒵 is complete. This means that 
    every ρ in the upper half-plane of height ≤η 
     is on the critical line, is simple, and belongs to some subinterval in the set 𝒵. 
     



There are important cases where the subintervals in 𝒵 
should be allowed to appear with multiplicity. In such cases, 
the conclusions about the simplicity of zeros in parts (ii) and (iv–v) should be modified 
so as to account for
any nonsimple zeros already present in 𝒵.
For example, the L-function of an elliptic curve with analytic rank >1 has by definition 
a zero at s=1/2 of multiplicity >1. 
Note that in this case, part (iv) of the theorem 
gives an unconditional upper bound on the analytic rank of the elliptic curve. 
Bober <cit.> gave a
method to bound the analytic rank of elliptic curve L-functions via the “explicit formula,” 
conditional on the RH for the corresponding L-function.



Let us explicitly note that if ϵ = ± 1, then ξ(1/2 + it) is even, so 
any zero at the central point s=1/2 has even multiplicity.
Thus, in this situation, it is unclear how the non-simple 
zero at s=1/2 can be detected rigorously by numerical means, via the intermediate value theorem, 
as there will be no sign change to detect. All this is to say that 
if ϵ = ± 1 and the zeros of height ≤τ 
have been sufficiently resolved, then the sum over intervals of the form [-γ_0,γ_0]∈𝒵 
is expected to be empty.


 


   We prove part (i). Let {ρ_1,…, ρ_m} be a set of m zeros ρ_j = β_j + iγ_j off of the critical line of height ≤η with (ρ_j) > 1/2 and (ρ_j) > 0, possibly with repetition up to multiplicity. For each ρ_j, there are necessarily 4 symmetric, distinct zeros ρ_j, 1-ρ_j, ρ_j, and 1-ρ_j. These 4 counterexample zeros will collectively contribute to w_1,δ a value of
   
    ϕ(β_j, γ_j, δ) + ϕ(β_j, -γ_j, δ) = 2ϕ(β_j,γ_j,δ).

   Using lemma <ref> and the monotonicity of ϕ(β,η,x) in η, we find that the minimum of the possible contribution from these 4 counterexample ρ_j is at least
   
    2·min_β∈ [0,1]ϕ(β,η,δ) = 2·min(ϕ(0,η,δ),ϕ(1/2,η,δ)) = f_1(η,δ,1).

   Note that this lower bound is independent of β_i and γ_i. Thus, the m counterexample zeros ρ_1,…, ρ_m along with their 3m symmetric zeros contribute at least m· f_1(η,δ,1) = f_1(η,δ, m) to the value of w_1,δ. Thus, if
   
    f_1(η,δ,m) + C(𝒵,δ) > w_1,δ,

   then we have a contradiction, so there are strictly fewer than 4m zeros ρ off the critical line of positive 
   height ≤η. In the case of m=1 this means the RH holds in the interval (0,η]. In the case of m=2, 
   there is at most one set of 4 symmetric, non-real ρ off the critical line of height ≤η and they must be simple.
   
   Next, we prove part (ii). Note that the minimum contribution to w_1,δ from a non-real zero ρ on the critical line of height ≤η together with its symmetric part 1-ρ is ϕ(1/2,η,δ). Similarly, the contribution from m such zeros on the critical line is at least m·ϕ(1/2,η,δ) = f_2(η,δ,m). Therefore, if
   
    f_2(η,δ,m) + C(𝒵,δ) > w_1,δ,

   then we have a contradiction, so there are strictly fewer than 2m zeros ρ on the critical line of positive height ≤η which are not accounted for in the subintervals of 𝒵. In the case of m=1 this means the intervals in 𝒵 account for all non-real ρ on the critical line of height ≤η. In the case of m=2, at most one pair of non-real ρ and 1-ρ on the critical line of height ≤η are not accounted for in 𝒵. Since each subinterval of 𝒵 contains a sign-change (so corresponds to a zero of odd multiplicity), this case implies that all non-real ρ on the critical line of height ≤η are simple (including the possible pair ρ and 1-ρ missed by 𝒵). 
   

   We prove part (iii). By Lemma <ref>, the minimum of the contribution to w_1,δ from a real zero ρ off the critical line and its symmetric zero 1-ρ is ϕ(1/2,0,δ). Thus, the contribution from m pairs of real zeros off the critical line {ρ_1,1-ρ_1,…, ρ_m,1-ρ_m} is at least m·ϕ(1/2,0,δ) = h_1(δ,m). So if 
   
    h_1(δ,m) + C(𝒵,δ) > w_1,δ,
 
   there are strictly fewer than m such pairs of real zeros, so fewer than 2m total real zeros off the critical line. 
   
   We prove part(iv). Each repetition of the zero ρ=1/2 (possibly none) 
   contributes 1/2·ϕ(1/2,0,δ) to w_1,δ. So, if the zero at s=1/2 has multiplicity m, then these zeros have total contribution m/2·ϕ(1/2,0,δ) = h_2(δ,m). By the same arguments thus far, if
   
    h_2(δ,m) + C(𝒵,δ) > w_1,δ,
 
   then the multiplicity of the zero at s=1/2 is strictly smaller than m. 
   In the case m=1, this means we have non-vanishing of L(s) on the real line. In the case m=2, and combined with part (iii), 
   any real zero must be at the central point s=1/2 and must be simple.

  Lastly, we prove part (v). Suppose 
  F(η,δ) + C(𝒵, δ) > w_1,δ.
  By the definition of F(η,δ), we have
       
    F(η,δ) ≤min{f_1(η,δ,1),f_2(η,δ,1),h_2(δ,1)}.

   Therefore, by part (i) of the theorem, all non-real ρ are on the critical line. By part (ii), all non-real ρ on the critical line belong to some subinterval in the set 𝒵 and are thus simple. By parts (iii) and (iv), L(s) is non-vanishing on the real line (except for possibly a simple zero at ρ=1/2 included in 𝒵). These three cases leave no room for zeros outside of the simple zeros within the subintervals in the set 𝒵. Thus, the list of zeros in 𝒵 account for all zeros of L(s) of height ≤η and they are all simple, i.e. the list 𝒵 is complete.






§ GENERALIZATION IN THE ZETA CASE


Let
g(s):=(s-1)ζ(s). So, g is an entire function. 
The series expansion of g at s=1 is given by

    g(s)=1+∑_j≥ 0(-1)^jλ_j/j!(s-1)^j+1,

where λ_1,λ_2,… are the Stieltjes constants.[For instance, 
λ_1=-0.07281584…, λ_2=-0.00969036…, λ_3 = 0.00205383…,
and so on.] For any complex number z such that ζ(1-z) 0, we may write

    log g(s-z)=∑_j≥ 0 c_j,z (s-1)^j,     c_j,z=1/j![d^j/ds^jlog g(s-z)]_s=1,

for s sufficiently close to 1.[If z=0, then the coefficients c_j:=c_j,0 can be calculated 
easily in terms of the λ_j's. For example, c_0=0, 
        c_1=λ_0,  c_2=-λ_0^2/2-λ_1, 
        c_3=λ_0^3/3+λ_0λ_1+λ_2/2….]


Let k be a positive integer. Let z=x+iy be a complex number such that x<1 and z does not 
coincide with any zero ρ of ξ(s). If k>1 then

    v_k,z:=∑_ρ1/(ρ-z)^k=(-1)^k-1[ψ_k-1(3/2-z/2)/2^k(k-1)!+k c_k,z].

If k=1, then there is an additional term of 

    -1/2logπ.



The proof is similar to that of Proposition <ref> 
except the coefficients c_j,z are defined differently than the 
analogous coefficients d_j,δ 
due to the pole of zeta.



When k=1, we have
    
    v_1,z= - 1/2logπ+
        1/2ψ_0(3-z/2)+
        g'(1-z)/g(1-z).



One can compute the g'(1-z)/g(1-z)
using the Euler–Maclaurin summation formula; see for example <cit.>. 
However, if x<0 is large enough, then the following simpler formula could suffice, 
and has the same proof as that for Lemma <ref>.


If z=x+iy and x< 0, then

    g'(1-z)/g(1-z) = -1/z-∑_k= 1^K Λ(k)/k^1-z + ℛ(K,x),

where ℛ(K,x) satisfies the same bound as in Lemma <ref> but with r=1 and δ=x.


Theorem <ref> is the main result in this section. Since the main interest in the case of zeta is at large heights, we expand about a complex number z=x+iy where y>0 is typically large. Therefore, 
 the advantage provided by the symmetry of the ρ's about the real axis is mostly lost. 


    Let z=x+iy be a complex number such that x≤ 0 and y>0. 
    Let τ be a real positive number such that 
    τ≤ y. Let 𝒵 be a set of nonempty disjoint subintervals [γ_-,γ_+]⊆ [y-τ,y+τ] such that ξ(1/2+it) has a sign change in each subinterval. 
    Suppose further that y does not belong to any of the subintervals in 𝒵.
     Define
     
    D(𝒵,z)   :=∑_[γ_-,γ_+]∈𝒵
    γ_->y1/2-x/(1/2-x)^2+(γ_+-y)^2+∑_[γ_-,γ_+]∈𝒵
    γ_+< y1/2-x/(1/2-x)^2+(y-γ_-)^2.

     Further, for any real η such that 0<η≤ y, and with ϕ as in (<ref>), define
    
    g_1(η,x)   :=min(ϕ(0,η,x),ϕ(1/2,η,x)), 
    
            g_2(η,x)   := ϕ(1/2,η,x), 
    
            g_3(η,x)    :=min(ϕ(0,η,x),1/2 ·ϕ(1/2,η,x)).

    Then, for any real positive number η such that η≤ y we have the following.
     
        
  (i) If
    g_1(η,x) +D(𝒵,z) >  (v_1,z),
    then all the ρ's with height in [y-η,y+η] are on the critical line.
    That is, the RH holds in the interval [y-η,y+η].
    
  (ii) If
    g_2(η,x) +D(𝒵,z) >  (v_1,z),
    then all the ρ's on the critical line with height in [y-η,y+η] are simple.
    
  (iii) If g_3(η,x) +D(𝒵,z) >  (v_1,z),
    then the list 𝒵 is complete. This means that  
    every ρ in the upper half-plane with height in [y-η,y+η]
     is on the critical line, is simple, and belongs to some subinterval in the set 𝒵. 
     


     Let us prove part (i). Suppose there is a counter-example ρ=β+iγ such that γ∈ [y-η,y+η]. Then 1-ρ is a counter-example distinct from ρ. The contribution of ρ and 1-ρ to (v_1,z) is ϕ(β,γ-y,x). Since |γ-y|≤η, it follows by Lemma <ref> that this contribution is at least g_1(η,x). Moreover, the zeros from the set 𝒵 already contribute at least D(𝒵,z) to (v_1,z). So, if the inequality in (i) holds, and considering that any remaining zeros will contribute a nonnegative amount to (v_1,z), then we obtain a contradiction. Hence, the counter-example ρ cannot exist.

     We prove part (ii). Suppose there is a nonsimple zero ρ=1/2+iγ of multiplicity m such that γ∈ [y-η,y+η]. If ρ is already in the set 𝒵, then m≥ 3, since the zeros in 𝒵 have odd multiplicity (as they correspond to sign changes of ξ(1/2+it)). If ρ is not in 𝒵, then m≥ 2. In either case, there are at least two zeros on the critical line with ordinates in [y-η,y+η] that are missing from 𝒵. So, arguing as in part (i) and using Lemma <ref>, the contribution of these missing zeros to (v_1,z) is at least g_2(η,x). So, if the inequality in (ii) holds, then we obtain a contradiction since any remaining zeros will contribute a nonnegative amount to (v_1,z). Hence, such a nonsimple  ρ cannot exist.

      Lastly, we prove part (iii). Note that g_1(η,x)≥ g_3(η,x) and g_2(η,x)≥ g_3(η,x). So, if the inequality in (iii) holds, then all the zeros with height in [y-η,y+η] are on the critical line and are simple. Thus, 
      in seeking a contradiction we may assume without loss of generality that there is a simple zero ρ=1/2+iγ such that γ∈ [y-η,y+η] and γ is not in any subinterval [γ_-,γ_+]∈𝒵. 
      But the contribution of such ρ to (v_1,z) is at least 1/2 ·ϕ(1/2,η,x). Hence, if the inequality in (iii) holds, then we obtain a contradiction, like before. So, such a missing ρ cannot exist.



By using the shift z=-1/2+i 14.1 in Theorem <ref> along with the 12 initial zeros of ζ(s), one can verify that ρ_1=1/2+iγ_1 and ρ_2=1/2+iγ_2 are the only zeta zeros with ordinates in the window [6.5360, 21.6640]. Since the value v_1= 0.0230957… that Riemann computed already tells us that there are no zeta zeros of height less than 6.56, this yields that ρ_1 and ρ_2 are indeed the first two zeta zeros. By comparison, verifying ρ_1 and ρ_2 are the first two zeta zeros using just the value v_1 requires accounting for the contribution of 52 initial zeros of zeta. 




§ IMPROVEMENTS


Instead of using nonnegativity to simply drop the contribution to (v_1,z) 
of the tail of the zeros sum, 
we derive  a lower bound on the contribution of the tail.
 Incorporating this into Theorem <ref>  greatly 
 improves the efficiency of our verification method 
 at large heights (i.e. when y is large). Hence, 
the RH can be verified via our method 
in a much wider window than before (i.e. for a much larger η). Specifically, whereas the basic verification method in Theorem <ref> is only expected to succeed 
in windows of size η≪√(τ/log y), the improved method in Theorem <ref> 
 is expected to succeed in windows of size η≫τ/√(log y).
 
In addition, we derive an upper bound on the contribution of the tail of the zeros sum. 
This can sometimes allow us to prove the incompleteness of a supplied 
list of zeros in a given range, as shown in Theorem <ref>.


Let z=x+iy be a complex number and τ be a real number. 
Suppose that x< 0 and 1-2x<τ< y.
For any real number c such that 168π<c<y-τ, we have
    
    - b(z,τ,c) ≤∑_ρ
     |γ-y|>τ1/ρ-z- 1-2x/2πτlogy/2π≤ B(z,τ,c),

    where
    
    b(z,τ,c):= 1/2π·[ϵ_11-2x/τ+ϵ_2+ϵ_3logy/2π]
    + ϵ_4+ϵ_5/2,
       B(z,τ,c):= 1/2π·[ϵ_11-2x/τ+ϵ_2]
    + ϵ_4+ϵ_5/2+ϵ_6,

and defining ℓ(u) and ℓ_1(u) as in (<ref>) we have

    ϵ_1(y,τ,c):=4π^2· 0.006·[1/(y+τ)^2+1/c^2],
       ϵ_2(z,c):=1-2x/2y·log2y/c,
       ϵ_3(z,τ,c):= [(1-x)^2/3τ^3+1/y-c]· (1-2x),
       ϵ_4(z,τ,c) := (2-4x/τ^2+2-4x/(y-c)^2)·ℓ(2y),
       ϵ_5(z,τ) := 4-8x/τ^3·ℓ_1(2y),
       ϵ_6(z,c) := 1-2x/2y·(2y-c)^2log(2y-c)-(y-c)^2log(y-c)/π (y-c)^2.






See <ref>.



    Let z =x+iy, τ, c, and the functions b(z,τ,c) and B(z,τ,c) all be given as in Proposition <ref>. Furthermore, let 𝒵 and the functions D(𝒵,z), ϕ(β,η,x), g_1(η,x), g_2(η,x), and g_3(η,x) be given as in Theorem <ref>. Define
    
    r(z,τ,c)    := 1-2x/2πτlogy/2π - b(z,τ,c)
    
            R(z,τ,c)    := 1-2x/2πτlogy/2π + B(z,τ,c)

    For any real positive number η such that η≤ y we have the following improvements to Theorem <ref>.
    
        
  (i) If
        g_1(η,x) + D(𝒵,z) + r(z,τ,c) >  (v_1,z),
        then all the ρ's with height in [y-η,y+η] are on the critical line. That is, the RH holds in [y-η,y+η].
        
  (ii) If
        g_2(η,x) +D(𝒵,z) + r(z,τ,c) >  (v_1,z),
        then all the ρ's on the critical line with height in [y-η,y+η] are simple.
        
  (iii) If g_3(η,x) +D(𝒵,z) + r(z,τ,c) >  (v_1,z),
        then every ρ in the upper half-plane with height in [y-η,y+η]
        is on the critical line, simple, and belongs to some subinterval in the set 𝒵. 
     
     In addition to these improvements, the upper bound in Proposition <ref> yields the following counterpart.
     
         
  (iv) If D(𝒵,z) + R(z,τ,c) <  (v_1,z), then 𝒵 does not account for all the ρ's with height in [y-τ,y+τ]. This means there is a subinterval in 𝒵 that contains the ordinates of at least three ρ's (including multiplicity), or there is ρ=1/2 + iγ such that γ∈ [y-τ,y+τ] and γ is not in any subinterval in 𝒵,  or there is ρ off the critical line with height in [y-τ,y+τ]. 
    


    Parts (i)–(iii) follow directly from the arguments in Theorem 10, except that these bounds account for the contribution from zeros ρ outside of the ordinate window [y-τ,y+τ] for which we have zeros data.

    For part (iv), if D(𝒵,z) + R(z,τ,c) <  (v_1,z), then there necessarily are zeros whose (positive) contribution to (v_1,z) is not being accounted for. More explicitly, since
    
    ∑_ρ
     |γ-y|>τ(1/ρ-z) ≤ R(z,τ,c),

    R(z,τ,c) already accounts for the maximum possible contribution from all zeros ρ with |γ - y| > τ. Therefore, any deficiency in contribution to (v_1,z) must arise from some ρ = β + iγ satisfying |γ - y|≤τ 
    that has not been already accounted for 
    in 𝒵.



    It is possible that a further small improvement would be made by incorporating explicit zeros-density estimates, in addition to the explicit bounds on S(u) and its integral that are already included. 
    
    
    
    
    




§ NUMERICAL EXAMPLES


The examples in this section are meant for illustration, to show how the method we described behaves 
in practice on representative examples.
The data in this section was obtained from <cit.> and <cit.>, 
and using LCALC <cit.> as well as SageMath <cit.>.
Our working assumption is that the zeros ordinates from <cit.> and <cit.> are accurate within ± 10^-10, and the zeros ordinates obtained using <cit.> and <cit.> 
are accurate to within ± 10^-8, though it is possible the accuracy is higher.
We used this assumption to determine the interval [γ_-,γ_+] corresponding to each zero ordinate γ.
Numerical calculations were done using the interval arithmetic package in mpmath <cit.>. We also used FLINT <cit.> to compute the polygamma function when no exact value was available. The code for the implementation is available as a GitHub repository <cit.>. 




 §.§ The Riemann zeta function
 
We used Theorem <ref> and Theorem <ref>
for verification using

    y = 10^28 + 501675.8,     x=-2,    τ = 501575.4,     c= y/2.

Our set 𝒵 was obtained from <cit.>. For z=x+iy, 
and with the aid of Corollary <ref> and Lemma <ref>, applied with K=10^7, we computed

    (v_1,z) ∈ [31.418062627034752, 31.418062627034846],


    D(𝒵,z) ∈ [31.417963253430945, 31.417963255019071],


    r(z,τ,c)∈ [0.000099372589781012325291744466523344471948495 ±  5*10^-45].

Based on this input data, Theorem <ref>, part(i), succeeded in verifying the RH 
for η=224, and Theorem <ref>, part (i), succeeded
verifying the RH for η=70216, which is much larger and contains 
1399910 zeros of the zeta function. Theorem <ref>, part (iii), also succeeded in 
verifying that completeness of 
the subset 

    𝒵∩ [y-η,y+η],    η=49650,
 
a window that contains 989881 zeros.
In the opposite direction, we applied Theorem <ref>, part (iv), to 
the subset 𝒵_0, which is the same as 𝒵 except 
the subinterval [γ_-,γ_+] corresponding to the ordinate
γ = 10^28 + 521738.816
was removed. We computed

    D(𝒵_0,z) ∈ [31.417963247220145, 31.417963248808271],


    R(z,τ,c)∈ [0.00009937291681087140202410471243137201884323 ± 10^-44].

Based on this input data, 
Theorem <ref> succeeded in proving that the set 𝒵_0 was indeed incomplete.



 §.§ Real Dirichlet L-function

Let d be a fundamental discriminant, 
χ_d be the corresponding real primitive character, and 
 L(s,χ_d) the corresponding Dirichlet L-function.
In the notation of <ref>, we have r=1, N=d, ϵ=1, and 
if d<0 then μ_1=1. 
We applied Theorem <ref>, part (i),
to verify the RH using

    d=-1159523,    δ = -1,    τ = 1692.8.
 
The coefficients arising from the Euler product are given by
α_p,1=χ_d(p).
Our set 𝒵 was obtained using <cit.>. 
With the aid of Corollary <ref> 
as well as Lemma <ref> applied with K=10^5 
we computed

    w_1,δ∈ [6.4702225452, 6.4702573982],
 

    C(𝒵,δ) ∈ [6.4644405451, 6.4644405588].

Based on this input data, Theorem <ref> succeeded in verifying the RH 
for L(s,χ_d) for η=32, a window containing 74 zeros with nonnegative ordinates.



 §.§ The Ramanujan τ L-function

Let τ be the Ramanujan tau function[So, τ(1) =1, τ(2) = -24, τ(3) = 252, τ(4) = -1472,….], 
and let L(s) be the Ramanujan tau L-function.[Therefore, 
L(s) is given by the Dirichlet series
L(s) = 1 +a_2 2^-s+ a_3 3^-s+a_4 4^-s+⋯ where a_n=τ(n) n^-11/2,
at least when σ > 1/2.] 
In the notation of <ref>, we have r=2, N=1, ϵ=1, μ_1=11/2, and μ_2=13/2.
We applied Theorem <ref>
to verify the RH using

    δ = -1,    τ = 9877.3.
 
The coefficients α_1,p and α_2,p arising from the Euler product for L(s) 
are given by the roots of the polynomial x^2-τ(p)p^-11/2x + 1.  
Our set 𝒵 was obtained using <cit.> and <cit.>. 
With the aid of Corollary <ref> 
and Lemma <ref>, applied with K=10^5, 
we computed

    w_1,δ∈ [0.1671717623, 0.1672414682],
 

    C(𝒵,δ) ∈ [0.1663983945, 0.1663983946].

Based on this input data, Theorem <ref> succeeded in verifying the RH for L(s)
for η=84, a window which includes 46 zeros with nonnegative ordinates.



 §.§ Elliptic curve L-function

Let E be an elliptic curve over ℚ of conductor N=Δ_E.
Let L(s,E) be the corresponding elliptic curve L-function.
In the notation of <ref>, we have r=2, N=Δ_E, ϵ= 1 or ϵ=i, 
μ_1=1/2, and μ_2=3/2.
We applied Theorem <ref> with

    δ = -1,    τ = 90,
 
to verify the RH for the elliptic curve E with 
minimal Weierstrass equation

    E: y^2+y=x^3-x.

According to <cit.>, E has conductor 37 so that N=37, 
and the sign of the functional equation of L(s,E) is -1 so that, 
in the notation of <ref>, ϵ= i. 
To calculate the Euler factors of L(s,E), let
|E(𝔽_p)| denote 
the number of solutions (x,y)∈𝔽_p×𝔽_p
that satisfy the minimal Weierstrass equation (<ref>)
together with the point at infinity that lies on E. Define

    b(p):=p+1-|E(𝔽_p)|.

Then
the coefficients α_1,p and α_2,p arising from the Euler product for L(s,E) 
are the roots of the polynomial x^2-b(p)p^-1/2x + 1, provided p 37.
If p=37, then α_p,1=-1/√(p) and α_p,2=0.
Our set 𝒵 was obtained from <cit.>. 
With the aid of Corollary <ref> 
and Lemma <ref>, applied with K=10^5, 
we computed

    w_1,δ∈ [1.2186382841, 1.21870798992],
 

    C(𝒵,δ) ∈ [1.160632197991927, 1.160632199964985].

Based on this input data, Theorem <ref> succeeded in verifying the RH for L(s,E)
for η=10, a window which contains 5 zeros with nonnegative ordinates.




§ PROOF OF PROPOSITION <REF>

Recall the function ϕ defined in (<ref>). 
For ρ = β+iγ, we have

    [1/ρ-z+ 1/1-ρ-z] = ϕ(β,γ-y,x).

Also, if |γ-y|>τ>1-x, then Lemma <ref>, parts (ii–iii) give

    ϕ(0,γ-y,x)≤ϕ(β,γ-y,x) ≤ϕ(1/2,γ-y,x).

Now, for u≥ 0 let N(u) be the number of zeta zeros with ordinates in [0,u], and extend N(u) to u<0 by requiring it to be odd. Using Stieltjes integrals, we define

    L(z,τ)   :=∫_|u-y|>τϕ(0,u-y,x) dN(u)/2,
    
        U(z,τ)   := ∫_|u-y|>τϕ(1/2,u-y,x) dN(u)/2.

The double inequality (<ref>) thus gives

    L(z,τ)≤∑_ρ
     |γ-y|>τ 1/ρ-z≤ U(z,τ).

We bound L and U from below and above, respectively,  
starting with L. 

To this end, since the integrand in L is nonnegative, a lower bound on L
can be obtained by restricting the integration interval to τ<|u-y|<y-c. 
Doing so, followed by the change of variable u← u-y, gives

    L(z,τ) ≥1/2·∫_τ<u<y-cϕ(0,u,x) d[N(y+u)-N(y-u)].

On the other hand, it is known <cit.> that 

    N(u) = 1/πθ(u)+1+S(u),

where θ(u)=  [π^-iu/2Γ(1/4+iu/2)] and,
if u does not coincide with the ordinate of 
any nontrivial zero, S(u)=π^-1ζ(1/2+iu).[The arguments are defined by a continuous variation starting at s=2, going up vertically to s=2+iu, and then horizontally to s=1/2+iu.] Also, θ(u) is a smooth odd function and S(u) is right-continuous with 
jump discontinuities at the zeros ordinates. Thus, combining (<ref>) and (<ref>), and defining

    L_θ(z,τ)   :=∫_τ<u<y-cϕ(0,u,x) (θ'(y+u)+θ'(y-u)) du,
     
        L_S(z,τ)   :=∫_τ<u<y-cϕ(0,u,x) d[S(y+u)-S(y-u)],

where θ' is the derivative of θ with respect to u, 
we obtain

    L(z,τ) ≥1/2π L_θ(z,τ)- 1/2 |L_S(z,τ)|,


We first bound L_θ from below.
By <cit.>, if u>0, then 

    |θ'(u)- 1/2logu/2π|≤4π^2· 0.006/u^2.

So, in view of (<ref>), we are led to consider

    1/2logy+u/2π+1/2logy-u/2π= logy/2π +1/2log(1-u^2/y^2).

Expanding the right-side in (<ref>) about u=0, and using the Lagrange form of the remainder, as well as (<ref>), we see that for τ< u< y-c, 

    |θ'(y+u)+θ'(y-u) -logy/2π|≤u^2/y^2-u^2+ϵ_1(y,τ,c),

where

    ϵ_1(y,τ,c)=4π^2· 0.006·[1/(y+τ)^2+1/c^2].


On the other hand, substituting the following simple bound into the integral in (<ref>) below,

    0≤ϕ(0,u,x) ≤1-2x/u^2,
 
and evaluating the resulting integral in closed-form gives the inequality

    0≤∫_τ^y-cϕ(0,u,x)·u^2/y^2-u^2 du ≤ϵ_2(z,c),

where

    ϵ_2(z,c)=1-2x/2y·log2y/c.

Additionally, using the anti-derivative formula

    ∫ϕ(0,u,x) du = arctan(u/1-x)-arctan(u/x),

together with the following double inequality (from the Laurent series for arctan), which is valid for u>τ>1-x,

    0 ≤[arctan(u/1-x)-arctan(u/x)]-[π - 1-2x/u]
    ≤(1-x)^3-x^3/3u^3,

we obtain

    ∫_τ^y-cϕ(0,u,x) du ≥1-2x/τ- ϵ_3(z,τ,c),

where, after using the elementary inequality (1-x)^3-x^3<(1-x)^2(1-2x),

    ϵ_3(z,τ,c)= [(1-x)^2/3τ^3+1/y-c]· (1-2x).

Therefore, combining (<ref>), (<ref>), (<ref>),
and (<ref>), we obtain

    L_θ(z,τ) ≥(1-2x/τ-ϵ_3) (logy/2π-ϵ_1 )-ϵ_2.


We now calculate an upper bound on L_S, which is defined in (<ref>). 
Let ϕ' denote the derivative of ϕ with respect to u. 
Using integration by parts and Lemma <ref>, part (iv), together with the intermediate value theorem, we obtain

    |L_S(z,τ)|   ≤  2 [ϕ(0,τ,x)+ϕ(0,y-c,x)]·sup_c<u<2y|S(u)|
          -2 ϕ'(0,τ,x)·sup_c<u_1<u_2<2y|∫_u_1^u_2S(u) du|.

If c> e, then <cit.> gives the bound |S(u)| ≤ℓ(u). And 
if c>168π, then <cit.> gives the bound |∫_u_0^uS(t) t|≤ℓ_1(u). 
So, using the simple bound (<ref>) for ϕ as well as the bound

    -2+4x/u^3≤ϕ'(β,u,x)≤ 0,

valid for 0≤β≤ 1, we obtain

    |L_S(z,τ)|≤ϵ_4(z,τ,c) + ϵ_5(z,τ),

where 

    ϵ_4(z,τ,c) = (2-4x/τ^2+2-4x/(y-c)^2)·ℓ(2y), 
       ϵ_5(z,τ) = 4-8x/τ^3·ℓ_1(2y).

Combining (<ref>), (<ref>), (<ref>) yields
the lower bound in the proposition.

To derive the upper bound in the proposition, we bound U(z,τ) in (<ref>) from above. 
Let us write U(z,τ) = I_1(z,τ)+I_2(z,τ) where

    I_1(z,τ)   := ∫_τ<|u-y|≤ y-cϕ(1/2,u-y,x) dN(u)/2,
    
        I_2(z,τ)    :=  ∫_|u-y|> y-cϕ(1/2,u-y,x) dN(u)/2.

I_1 is estimated by an analogous calculation to that used for L(z,τ).
The difference is that the formula (<ref>) and the double inequality (<ref>)  are replaced with
 the formula

    ∫ϕ(1/2,u,x) du = 2arctan(2u/1-2x),

and the following double inequality, valid for u>τ>1-2x,

    0 ≤    2arctan(2u/1-2x)-[π - 1-2x/u]
    ≤(1-2x)^3/12u^3.

Consequently,  the formula (<ref>) is replaced with

    ∫_τ^y-cϕ(1/2,u,x) du ≤1-2x/τ.

So that the term ϵ_3 in (<ref>) may be replaced with zero. 
Also, since we are looking for an upper bound, the - signs in (<ref>) should be replaced 
with + signs. Put together,

    I_1(z,τ) ≤1-2x/2πτ(logy/2π+ϵ_1 )+ϵ_2/2π+ϵ_4+ϵ_5/2.


Next, we bound I_2. After  the change of variable u← u-y we obtain

    I_2(z,τ) = 1/2∫_y-c^∞ϕ(1/2,u,x) dN(y+u) - 
        1/2∫_y-c^∞ϕ(1/2,u,x) dN(y-u).

Using integration by parts together with the observation ϕ(1/2,u,x)≪ 1/u^2 
and the facts that N(u)≪ ulog u and non-decreasing, we obtain

    I_2(z,τ) ≤∫_y-c^∞|ϕ'(1/2,u,x)| N(y+u) du.

Therefore, on substituting the bound on ϕ' given in (<ref>) and the bound 

    |N(u)|≤u/2πlogu/2π,      (u≥γ_1)
 
which follows from e.g. <cit.>, we obtain after a small calculation

    I_2(z,τ)≤ϵ_6(z,c),

where ϵ_6(z,c) is  defined as in the statement of the proposition. 
The claimed upper bound then follows on combining this with (<ref>).



§ PROOF OF LEMMA <REF>


Proof of parts (i)–(iii): Taking the partial derivative of ϕ with respect to β, we find with the aid of a computer algebra system that

    ∂/∂βϕ(β,η,x) = 
        -(2β - 1)(2x-1)G(β,η,x)/[((β-x)^2+η^2)((1-β-x)^2 + η^2)]^2,

where G is a degree 4 monic polynomial in β,

    G(β,η,x) :=    β^4 - 2β^3 + (1 - 2η^2 + 2x - 2x^2)β^2 + 2(η^2 - x + x^2)β
       + η^2(2x - 2x^2 - 1 - 3η^2) + x^2(1 - 2x + x^2).

satisfying G(β,η,x)=G(1-β,η,x).
Note that the sign of the partial derivative of ϕ with respect to β 
is the same as the sign of (2β-1)G(β,η,x).

We have the formulas

    G(β,0,x)    = (β-x)^2(1-β-x)^2, 
    
    
     ∂/∂β G(β,η,x)    = 2(2β-1) (β-r_+)(β-r_-),

where 

    r_± := 1 ±√(4η^2 + (2x-1)^2)/2    so that     r_- < 0  and  1 < r_+,

as well as the formulas

    [∂^2/∂β^2 G(β,η,x) ]_β=1/2   =
        -4η^2 - (2x-1)^2,
    [∂^2/∂β^2 G(β,η,x) ]_β=r_±   =
        8η^2 + 2 (2x-1)^2.












    
    ∂/∂βϕ(β,η,x) := ϕ_β(β,η,x)
    
            =    η^2 - (β-x)^2/((β-x)^2 + η^2)^2 - η^2 - (1-β-x)^2/((1-β-x)^2 + η^2)^2
    
            =    (η^2 - (β-x)^2)((1-β-x)^2 + η^2)^2 - (η^2 - (1-β - x)^2) ((β-x)^2 + η^2)^2/[((β-x)^2+η^2)((1-β-x)^2 + η^2)]^2
    
            =    -(2β - 1)(2x-1)G(β,η,x)/[((β-x)^2+η^2)((1-β-x)^2 + η^2)]^2

        where
    
    G(β,η,x) := β^4 - 2β^3 + (1 - 2η^2 + 2x - 2x^2)β^2 + 2(η^2 - x + x^2)β + η^2(2x - 2x^2 - 1 - 3η^2) + x^2(1 - 2x + x^2).

        Now, consider the following equations:
    
    G(β,0,x)    = β^4 - 2β^3 + (1 + 2x - 2x^2)β^2 + 2x(x - 1)β + x^2(x-1)^2
       = (β-x)^2(1-β-x)^2
    ∂/∂ηG(β,η,x)   = -4β^2η+4βη + 2(2x-2x^2-1)η - 12η^3
       = -η[12η^2 + (4β^2 - 4β + 1) + (4x^2 - 4x + 1)]
       = -η[12η^2 + (2β-1)^2 + (2x-1)^2]
    ∂/∂β G(β,η,x)    = 4β^3 - 6β^2 + 2(1 - 2η^2 + 2x - 2x^2)β + 2(η^2 - x + x^2)
       = 2(2β-1)(β^2-β - η^2 + x - x^2)
       = 2(2β-1)(β - (1 + √(1 + 4(η^2 + x^2 - x))/2))(β - (1 - √(1 + 4(η^2 + x^2 - x))/2))
       = 2(2β-1)(β - (1 + √(4η^2 + (2x-1)^2)/2))(β - (1 - √(4η^2 + (2x-1)^2)/2))
    ∂/∂β^2 G(1/2,η,x)    = 12(1/2)^2 - 12(1/2) + 2(1-2η^2 + 2x - 2x^2)
       = -4η^2 - 4x^2 + 4x - 1
       = -4η^2 - (2x-1)^2



    Taking (<ref>) as our “starting point" in some sense, and viewing it as a function of β, we see that
    G(β,0,x) has two local minima (and β-axis intercepts) at β = x and β = 1-x with a (positive) local maximum at β = 1/2. As η>0 increases, we see from (<ref>) and (<ref>) together with (<ref>)
     that the two local minima locations r_± move away from 1/2.
     In comparison, as follows from (<ref>) and (<ref>), 
     β = 1/2 remains a local maximum of G(β,η,x) (and a global maximum on the β-interval [0,1]),
     albeit with a monotonically decreasing value of G(1/2,η,x). The latter claim can be seen from the negativity of the partial derivative
     
    ∂/∂ηG(β,η,x) = -η(12η^2 + (2β-1)^2 + (2x-1)^2).

     
     We now consider three cases.


    Case (1): Suppose G(β,η,x) > 0 on the β-interval (0,1), so that, by
        considering the sign of ∂ϕ/∂β in (<ref>), we find
        β=1/2 is a local minimum of ϕ(β,η,x). 
        In evaluating this case, we note by the negativity of the partial derivative in (<ref>), 
        if η and x are such that β = 0  is a root of G(β,η,x), then β=0 cannot 
        be a root for any greater η (with the same x). Also, β=0 is a root if and only if 
        the constant term of the polynomial 
        G(β,η,x) in (<ref>) is 0, hence if and only if 
        3η^4 + (2x^2-2x+1)η^2 + x^2(-x^2+2x-1)=0. Solving for η and 
        recalling the discussion following (<ref>), we therefore
        find G(β,η,x) > 0 throughout β∈ (0,1) if and only if 
        
    η≤√(-2x^2+2x-1 + √(16x^4-32x^3+20x^2-4x+1)/6)=:v(x).

        For such η, ϕ(β,η,x) has no local extrema in the interval β∈ (0,1) 
        except at β=1/2 where it 
        has a local minimum. Thus, if 0≤β≤ 1, then ϕ(β,η,x) is minimized at β=1/2 in this case.


        Case (2): Suppose G(β,η,x) < 0 on the β-interval (0,1), 
        so that, by considering the sign of ∂ϕ/∂β in (<ref>), we find
        β=1/2 is a local maximum of ϕ(β,η,x). 
        On the other hand, G(β,η,x) maintains a negative sign throughout β∈ (0,1) if and only if 
        the local extremum of G(β,η,x) that occurs at β=1/2 is negative. 
        By direct calculation, we have
        
    G(1/2,η,x)=    1/16(1-2x)^4-1/16(1-2x)^4η^2-3η^4.

        Setting G(1/2,η,x) = 0 and solving for η, we find that G(β,η,x) < 0 throughout β∈ (0,1) if and only if 
    η > 1-2x/2√(3).

        
        
        
        
        
        
        
        
        
        
        
        
        
        For such η, ϕ(β,η,x) has no local extrema in the interval β∈ (0,1) except 
        at β=1/2 where it 
        has a local maximum. Thus, if 0≤β≤ 1, then 
        ϕ(β,η,x) is minimized at the boundary points β=0,1 and maximized at β=1/2 in this case.

        
        

        Case (3): Suppose G(β,η,x) does not maintain its sign throughtout the β-interval (0,1), so that
        G(β,η,x) has at least one root at some β∈(0,1). By the symmetry of G(β,η,x) about β=1/2
        as well as the discussion following (<ref>), there can be at most two such roots, one
        in the subinterval (0,1/2] and another symmetric root in the subinterval [1/2,1).
        By the work done thus far, such roots occur if and only if
        
    v(x) < η≤1-2x/2√(3).

        When these roots occur, they each correspond to local maxima of ϕ(β,η,x) as seen by considering the sign of ∂ϕ/∂β in (<ref>). Therefore in this case, ϕ(β,η,x) is minimized either at the boundary β = 0 (equivalently β = 1), or at the center β =1/2. To find the point of transition between these two situations, we set ϕ(0,η,x) = ϕ(1/2,η,x) and find that the transition 
        occurs when
        
    η = √(x(x-1)/3).

    
    
    
     Summary: From the above 3 cases, we have the following behavior of the minimum of 
        ϕ(β,η,x) for β∈ [0,1].
    
        
  (a) If η≤√(x(x-1)/3), then ϕ(β,η,x) is minimized at β=1/2
        
  (b) If η > √(x(x-1)/3), then ϕ(β,η,x) is minimized at β=0 (equivalently β=1).
    
    This covers claims (i) and (ii) in the statement of the lemma. Claim (iii) in the lemma 
    follows from Case (2) above.


Proof of part (iv):
    Writing ϕ'(β,u,x) as the partial derivative of ϕ(β,u,x) with respect to u and ϕ”(β,u,x) similarly. 
    We have
    
    ϕ'(β,u,x)    = 2(x-β)u/[(x-β)^2 + u^2]^2 + 2(x-(1-β))u/[(x-(1-β))^2 + u^2]^2.

    Since x≤ 0, β∈ [0,1], and u > 0, by (<ref>) we have that ϕ'(β,u,x) < 0 for any such β,u, and x.
    
    For the claims regarding the increasing nature of ϕ'(β,u,x), we begin by evaluating ϕ” at β = 1/2 and find
    
    ϕ”(1/2,u,x) = 2(1-2x)[3u^2 - (1/2 - x)^2]/[(1/2-x)^2 + u^2]^3.

    From this, we see that ϕ'(1/2,u,x) is increasing if and only if 
    
    u > 1/2 - x/√(3) = 1-2x/2√(3),

    yielding the first of these claims.

    Similarly, evaluating ϕ” at β=0 gives us
    
    ϕ”(0,u,x) = 2x(x^2-3u^2)/(x^2+u^2)^3 + 2(x-1)((x-1)^2-3u^2)/((x-1)^2+u^2)^3.

    The first term in (<ref>) is positive only when u > |x|/√(3) and the second term only when u > |x-1|/√(3). 
    So, since x≤ 0, ϕ'(0,u,x) is increasing when 
    
    u > 1-x/√(3) = 2-2x/2√(3).

    









§ CONCLUSIONS AND FUTURE DIRECTIONS

We presented a method to verify the RH for zeta at large heights and to verify the RH for a general class of L-functions at low heights. The method is simple to understand and implement and we demonstrated its efficacy on a variety of L-functions using interval arithmetic. We also presented a significant improvement to the method in the case of zeta by incorporating explicit bounds on S(t) and integrals of S(t).

In forthcoming work, we will develop and detail further generalizations of this verification method. These generalizations include, among other things, consideration of w_k,δ and v_k,z when  k>1, further improvements in the case of zeta at large heights, the special case of Dirichlet L-functions to real primitive characters, as well as the extending of the improvements in <ref> to 
a more general setting.



§ ACKNOWLEDGEMENTS

We are grateful to Georg-August-Universität Göttingen,
Niedersächsische Staats- und Universitätsbibliothek Göttingen, for helping us locate some of
Riemann's unpublished notes and giving us permission to reproduce them in this paper. 
Megan Kyi thanks the OH5-OSU SURE Undergraduate Research program at the Ohio State University, Columbus, for 
their support.








