A refined random matrix model for function field L-functions
Will Sawin
 

§ ABSTRACT  We propose a refinement of the random matrix model for a certain family of L-functions over 𝔽_q[u], using techniques that we hope will eventually apply to an arbitrary family of L-functions. This consists of a probability distribution on power series in q^-s which combines properties of the characteristic polynomials of Haar-random unitary matrices and random Euler products over 𝔽_q[u]. The support of our distribution is contained in the intersection of the supports of the two original distributions. The expectations of low-degree polynomials in the coefficients of our series approximate the expectations of the same polynomials in the coefficients of random Euler products, while the expectations of high-degree polynomials approximate the expectations of the same polynomials in the coefficients of the characteristic polynomials of random matrices. Furthermore, the expectations of absolute powers of our series approximate the <cit.>-<cit.> prediction for the moments of our family of L-functions. 





A refined random matrix model for function field L-functions
    Will Sawin
    September 9, 2024
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§ INTRODUCTION



We begin by defining two probability distributions: one describing uniformly random multiplicative functions and their associated Euler products, and the other uniform random matrices and their characteristic polynomials. We next construct a probability distribution as a hybrid of both, which describes non-uniform random matrices. We then state our results about this hybrid distribution, and explain how it can be used to model the behavior of a certain family of Dirichlet L-functions.

Let 𝔽_q[u]^+ be the set of monic polynomials in one variable over a finite field 𝔽_q. We say a polynomial in 𝔽_q[u]^+ is prime if it is irreducible. Take for each prime π∈𝔽_q[u]^+ an independent random variable ξ(π) uniformly distributed on the unit circle in ℂ and form the random Euler product

    L_ξ(s)= ∏_𝔭∈𝔽_q[u]^+ 
    prime1/1- ξ(𝔭) q^-s 𝔭 .

We can extend ξ uniquely to a function ξ𝔽_q[u]^+ →ℂ that is completely multiplicative in the sense that ξ(1)=1, ξ(fg)=ξ(f)ξ(g) for all f,g∈𝔽_q[u]^+. In other words, ξ is a Steinhaus random multiplicative function. Then we can equally well express L_ξ(s) as a sum ∑_f ∈𝔽_q[u]^+ξ(f) q^- s  f. We have

    log  L_ξ(s)=∑_𝔭∈𝔽_q[u]^+ 
    primelog1/1- ξ(𝔭) q^-s 𝔭 = ∑_𝔭∈𝔽_q[u]^+ 
    prime∑_m=1^∞ξ(𝔭)^m q^ - s m 𝔭/m   .

Let X_n ,ξ be the coefficient of q^- ns in log L_ξ(s), i.e.  
    X_n,ξ = ∑_d | n∑_𝔭∈𝔽_q[u]^+ 
    𝔭 =d  
    primed/nξ(𝔭)^n/d .


Fix k a natural number. Assume q>2.  Let F(x_1,…, x_k) be the probability density function of the tuple of random variables X_1,ξ,…, X_k,ξ. (We assume q>2 since it is not hard to check that this probability density function does not exist for q=2 as long as k≥ 2.)

Let  be the set of power series in q^-s with constant coefficient 1. The power series L_ξ(s) lies in . We endow  with a topology by viewing it as a product of copies of ℂ and taking the product topology, and consider the Borel Σ-algebra. Let μ_ep be the measure on  given by the distribution of the random variable L_ξ.

For N a natural number and M in the unitary group U(N), let 
    L_M(s) =  ( I - q^1/2 - s  M).
 We have L_M∈. (In fact, L_M is a polynomial in q^-s and not just a power series.) Let μ_rm be the measure on  given by the distribution of the random variable L_M for M Haar-random in U(N). In other words, μ_rm is the pushforward of the Haar measure μ_Haar from U(N) to .

The probability distributions μ_ep and μ_rm can both be used as models for properties of random Dirichlet L-functions over 𝔽_q[u]. We now describe a distribution that combines properties of both and thus, we hope, serves as a better model than either.

Fix β∈( 1/4,  1/2) and N a natural number and let k= ⌊ N^β⌋. 


Consider the non-uniform measure on U(N)

    μ_weighted = γ F( - q^1/2(M),…, -q^k/2 (M^k)/k)/∏_j=1^k  ( e^  -   (M^j)^2 / j  j / q^j π) μ_Haar
 where γ is the unique constant that makes μ_weighted a probability measure.  We let μ_ch be the distribution of L_M for M a matrix in U(N) distributed according to μ_weighted, i.e. the pushforward of μ_weighted from U(N) to .

We think of μ_ch as a chimera in the sense of a strange hybrid of the more familiar creatures μ_ep and μ_rm. The measure μ_ch exists to serve as a model of a function field analogue of the Riemann zeta function, and hence combines the Euler product and random matrix perspectives on the zeta function. (We will meet the exact function field analogue which μ_ch models shortly in <ref>.) We prove three fundamental results about μ_ch describing the support of the measure and its integrals of the measure against a general class of test functions. These results explain which properties μ_ch shares with each of the simpler measures μ_ep and μ_rm. Combining these, we will in Theorem <ref> evaluate the integral against a specific test function that models the moments of the zeta function, and the resulting formula will look similar to predictions from <cit.> for the moments of the zeta function. 



Note that all our measures depend implicitly on the parameters q,N (except that μ_ep is independent of N), so it should not be surprising when error terms in our estimates for them depend on q or N. All implicit constants in big O notation will be independent of q,N except where noted.



 For N larger than some absolute constant, the support of μ_ch is contained in the intersection of the supports of μ_ep and μ_rm. 

Natural test functions to use on  are polynomials in the coefficients of the power series and their complex conjugates, i.e. we consider elements of the polynomial ring ℂ[ c_1,c_2,…, c_1, c_2,…] as functions on an element 1+∑_d=1^∞ c_d q^-ds of . We define the (weighted) degree of a polynomial in  ℂ[ c_1,c_2,…, c_1, c_2,…] by letting c_d and c_d have degree d for all d. We define the L^2 norm of such a polynomial using the random matrix measure, as 

    ϕ_2^2=∫_ϕ^2   μ_rm   =  ∫_U(N)ϕ ( L_M) ^2   μ_Haar .



 Assume that q>5. Let ϕ∈ℂ[ c_1,c_2,…, c_1, c_2,…] have degree ≤ k. For N sufficiently large in terms of β, we have

    ∫_ϕμ_ch  =  ∫_ϕμ_ep + O (e^ -(1/2 - o_N(1)) N^1-βlog (N^1-β)ϕ_2 ) .




 Assume that q>11. Let ϕ∈ℂ[ c_1,c_2,…, c_1, c_2,…]. Assume that for all polynomials ψ∈ℂ[ c_1,c_2,…, c_1, c_2,…] of degree ≤ k we have

    ∫_U(N)ϕ(L_M) ψ(L_M)μ_Haar =0 .

Then for N sufficiently large in terms of β, we have

    ∫_ϕμ_ch  =O_q ( N^-  βq-2/4ϕ_2   ) .



In interpreting <ref>, it is helpful to consider the heuristic that the typical size of ϕ on the support of μ_rm is approximately ϕ_2, and therefore, we should expect a trivial bound for the integral ∫_ϕμ_ch to be of size roughly ϕ_2. From this point of view we can view the factors of e^- (1/2 - o_N(1)) N^1-βlog (N^1-β) in <ref> and N^- βq-2/4 in <ref> as the amount of savings over the trivial bound, although obtaining an error term of size O( ϕ_2) is not completely trivial.



 §.§ A family of L-functions and their moments


We next explain how μ_ch can be used as a model for a certain family of L-functions. We first consider a family of characters (discussed in more detail in <cit.>). 

 We say a Dirichlet character ν( 𝔽_q[x]/x^N+2)^×→ℂ^× is “primitive" if ν is nontrivial on elements congruent to 1 mod x^N+1, and “even" if ν is trivial on 𝔽_q^×. For a Dirichlet character ν, define a function χ on monic polynomials in 𝔽_q[u] by, for f monic of degree d, 
    χ(f) = ν ( f(x^-1) x^d ) .
 It is easy to see that χ depends only on the N+2 leading terms of f. Let S_N+1,q be the set of characters χ arising from primitive even Dirichlet characters ν in this way. Because there are q^N+1 even Dirichlet characters of which q^N are imprimitive, S_N+1,q	 has cardinality q^N+1- q^N. 

For χ∈ S_N+1,q, form  the associated L-function 
    L(s,χ) = ∑_ f ∈𝔽_q[u]^+ χ(f)  f^-s.
 


The family of L-functions we consider consists of, for χ∈ S_N+1,q and t∈ [ 0,2π/log q ], the function L(s+it,χ). A random L-function of this family is obtained by choosing χ and t independently uniformly at random. Let us see why these L-functions form a reasonable model for the statistics of the Riemann zeta function. 

L(s+it,χ) is the Dirichlet series with coefficients f ↦χ(f) f^-it. Viewed as characters of the idele class group of 𝔽_q(u), these comprise all the unitary characters ramified only at ∞ with conductor exponent N+2 at ∞. They are thus comparable to the characters n → n^it of ℕ, which are the unitary characters of the idele class group of ℚ ramified only at ∞. Said in a more elementary fashion, n^it for t≤ T may be accurately approximated given the leading ≈log T digits of n as well as the total number of digits, and all multiplicative functions with this approximation property have the form n^it, while χ(f) f^-it may be computed exactly given the leading N+2 coefficients of f as well as the degree of f, and all multiplicative functions with this property (or even those that may be approximated given this information) are of the form χ(f) f^-it for some χ∈ S_N' +1,q for some N' ≤ N. Thus the statistics of L(s+it, χ) for random χ∈ S_N+1,q, t∈ [0, 2π/log q] are comparable to the statistics of ζ(s+it) for random t∈ [T,2T], i.e. the local statistics of the Riemann zeta function on the critical line.

The distribution of the coefficients  f ↦χ(f) f^-it converges in the large N limit to the distribution of a random multiplicative function ξ. (Without the average over t, they would converge to random multiplicative function subject to the restriction ξ(u)=1.)  On the other hand, by work of Katz <cit.>, in the large q limit the distribution of the L-functions L(s+it,χ) converges to the distribution μ_rm, as long as N ≥ 3. (Technically, we must express our power series in the variable q^1/2-s instead of q^-s for this convergence to make sense, as otherwise μ_rm depends on q.)  More precisely, <cit.> proves equidistribution of conjugacy classes in PU(N) whose characteristic polynomials correspond to L(s,χ) against the Haar measure of PU(N), and the additional averaging over t is equivalent to averaging over the fibers of U(N)→ PU(N).

Because of this N→∞ and q→∞ limiting behavior, the distribution of the family of L-functions L(s+it,χ) for finite q,N is expected to have some similarity with μ_ep and some with μ_rm. Thus μ_ch, which interpolates between μ_ep and μ_rm, is a plausible model for the distribution of L(s+it,χ). To test this model we must compare to facts known or expected to hold for the family of L-functions. We begin that investigation in this paper by comparing to the Conrey-Farmer-Keating-Rubinstein-Snaith predictions <cit.>, adapted to function fields by Andrade and Keating <cit.>, for moments of L-functions. The moment of L-functions we consider is 

    1/ q^N (q-1) log q/2π∑_χ∈ S_N+1,q∫_0^2π/log q∏_j=1^ L(s_j+i t ,χ) ∏_j=+1^ + L ( s_j+ it  ,χ)

i.e. the average over this family of the product of  special values of L(s+it,χ) with  special values of L(s+it,χ).  The recipe of <cit.> predicts a main term for this moment of

    MT_N^, (s_1,…, s_+ )= ∏_j=+1^+ q^ - (1/2-s_j) N∑_  S ⊆{1,…,+}
    S=∏_j ∈ S q^ (1/2-s_j) N∑_ f_1,…, f_+∈𝔽_q[u]^+ 
    ∏_j ∈ S f_j = ∏_j∉ S f_j∏_j∈ Sf_j ^ -1+ s_j ∏_j∉ Sf_j ^ -s_j .

In other words, in its most optimistic form the prediction is

    1/ q^N (q-1) log q/2π∑_χ∈ S_N+1,q∫_0^2π/log q∏_j=1^ L(s_j+i t ,χ) ∏_j=+1^ + L ( s_j+ it  ,χ)  = MT_N^, (s_1,…, s_+ ) +  O_q, ,,ϵ (  (q^N)^ - ( 1/2-ϵ))

and less optimistically one makes the same prediction with a larger error term.



Assume that q>11. Let  and  be nonnegative integers and s_1,…, s_ + be complex numbers with real part 1/2. Then

    ∫_( ∏_j=1^ L(s_j) ∏_j=+1^ + L ( s_j) )μ_ch   = MT_N^, (s_1,…, s_+ ) 
    + O_q,,(  N^ (+ )^2/2 -  βq-2/4).


For the integral on the left hand side of (<ref>), L should be understood as the variable of integration, i.e. L(s_j) is the function that takes a power series to its value at s_j, defined on the subset of  of power series in q^-s with radius of convergence > q^-1/2. The integral is well-defined since μ_ch is supported on the even smaller subset consisting of polynomials in q^-s of degree N. 

If we believe the CFKRS prediction for the moments, then <ref> implies that μ_ch has the same moments as the family of Dirichlet characters, up to a certain error, and thus gives evidence that μ_ch is a good model for the L-functions of the Dirichlet characters S_N+1,q (with additional averaging in the imaginary axis). Alternately, <ref> could be seen as giving a probabilistic explanation of the CFKRS prediction.

In interpreting <ref>, it is helpful to note that the main term MT_N^, (s_1,…, s_+ ) is, in the special case s _1=… = s_ +, a polynomial in N of degree . So the error term in (<ref>) is smaller than the main term by a factor of N^βq-2/4 - ^2 + ^2/2. In particular, it is actually smaller if q> 2 +  2 ^2 + 2 ^2/β and the number of coefficients of the polynomial that are visible in this estimate (in the sense that their contribution to the main term is greater than the size of the error term) is 
    min( ⌈βq-2/4 - ^2 + ^2/2⌉ ,  +1 ) .
 Thus for q sufficiently large depending on ,, all coefficients of the polynomial are visible in this sense.

We conjecture that an even stronger statement holds:

Let  and  be nonnegative integers,q>2 a prime power, s,…, s_ + complex numbers with real part 1/2, and A a real number.

    ∫_( ∏_j=1^ L(s_j ) ∏_j=+1^ + L ( s_j ) )μ_ch =  MT_N^, (s_1,…, s_+ ) + O_q,,,A(  N^ -A ).
 

If <ref> is true then the measure μ_ch correctly predicts every coefficient of the polynomial CFKRS main term. 

While this paper considers a particular family of L-functions, we hope that similar methods can be applied to essentially any family of L-functions, at least in the function field context. One just needs to consider a random Euler product whose local factors match the distribution of the local factors of the family of L-function (e.g. for the family of quadratic Dirichlet characters with prime modulus, take the Dirichlet series of random ± 1-valued completely multiplicative functions) and, if necessary, depending on the symmetry type, replace U(N) with O(N), SO(N), or Sp(N). Passing from random Euler products with continuous distributions to discrete distributions introduces some difficulties, but most likely not insurmountable ones. 





 §.§ Prior work


The oldest probabilistic model for the Riemann zeta function is the random Euler product ∏_p1/1- ξ(p) p^-σ  where ξ(p) are independent and identically distributed on the unit circle. The distribution of this random Euler product was proven by Bohr and Jessen <cit.> to give the limiting distribution of ζ(σ+it) for fixed σ >1/2, and Bagchi <cit.> proved a generalization giving the distribution of ζ(s+it) as a holomorphic function on a fixed domain to the right of the critical line. 

On the critical line, Selberg's central limit theorem shows that logζ(1/2+it)/loglog T has a Gaussian limiting distribution for t ∈ [T,2T] as T→∞. The division by loglog T means that this result is not sensitive to the exact size of zeta, and it similarly gives no information about the zeroes. Probabilistic models for zeta and L-functions that give precise descriptions of the behavior on the critical line must, for now, be conjectural.

A crucial starting point is the work of Montgomery <cit.>, who conjectured that the statistics of k-tuples of zeroes of the zeta function, in the limit over larger and larger intervals in the critical line, match the statistics of k-tuples of eigenvalues of a Haar-random matrix, in the limit of larger and larger random matrices, for each k, and provided evidence fo this.

Katz and Sarnak <cit.> observed that zeta and L-functions in the function field context arise from characteristic polynomials of unitary matrices of fixed size (depending on the conductor of the L-function) and that in the large q limit these unitary matrices become Haar-random for several natural families of L-functions, so in fact all statistics match statistics of random matrices in the large q limit. Using this, they made conjectures about the distribution of the low-lying zeroes of L-functions.

Keating and Snaith <cit.> used a random matrix model to study values of zeta and L-functions on the critical line, and not just their zeroes. In particular, they calculated the moments of the characteristic polynomial of a Haar-random unitary matrix at a point on the unit circle, in terms of the size of the matrix. To obtain a conjectural expression for the moments of the zeta function at a random point on the critical line, one has to substitute log T for the size of the matrix in this formula and then multiply by an arithmetic factor that expresses the contribution of small primes. Thus, if one models the values of the zeta function on a random strip of the critical line by the characteristic polynomial of a random unitary matrix on a strip of the unit circle, one obtains predictions for the moments that are conjecturally correct to within a multiplicative factor.

The situation was improved by Gonek, Hughes, and Keating <cit.>, using an Euler-Hadamard product, which expresses the zeta function locally as a product of one factor which roughly consists of the Euler factors at small primes and another factor which roughly consists of the contributions of nearby zeroes to the Hadamard product. (This depends on an auxiliary parameter – the more primes one includes, the fewer zeroes are needed, and vice versa). This thereby suggests a model where the first factor is modeled by the Euler product over small primes of a uniformly random multiplicative function and the second factor is modeled by the contributions of zeroes near a given point to the characteristic polynomial of a random unitary matrix, with the two factors treated as independent. (They were later proved to be asymptotically independent, conditional on the Riemann hypothesis, by Heap <cit.>.) For the moments, the <cit.> model recovers the same prediction as <cit.> of the product of a random matrix factor and an arithmetic factor.

A related but distinct approach to the moments of zeta and L-functions is the work of Conrey, Farmer, Keating, Rubinstein, and Snaith <cit.>.  This work did not directly predict the moments using a characteristic polynomial. Instead the authors found a particular formula for the (shifted) moments of the characteristic polynomial of a random unitary matrix and conjectured a formally similar formula for the (shifted) moments of the Riemann zeta function or another L-function, roughly speaking by inserting suitable arithmetic factors at an intermediate step in the calculation instead of at the end. However, the intermediate stages of their recipe lack a clear number-theoretic or probabilistic interpretation. In particular, it is not even obvious that their predictions for expectations of powers of absolute values of the zeta function are positive – this has to be checked separately. It is not clear that there exists any random holomorphic function whose moments are given by the <cit.> predictions for moments of zeta, though conjecturally a random shift of the Riemann zeta function would be an example.

Another approach to predicting moments of L-functions is by multiple Dirichlet series, initiated in the work of Diaconu, Goldfeld, and Hoffstein <cit.>. The highest-order terms in these predictions, made around the same time, agree with <cit.>, but the multiple Dirichlet series can be used to predict additional lower-order terms for certain families of L-functions, as in the work of Diaconu and Twiss <cit.>. Again these predictions are not probabilistic in nature, instead based on assuming that meromorphic functions defined by certain complicated multivariable sums have the greatest amount of analytic continuation allowed by their symmetry properties.

The predictions of <cit.> for moments of zeta on the critical line are polynomials in log T. The leading term of these moments agrees with the leading term originally predicted in <cit.> and probabilistically modeled by <cit.>. Thus, the prediction of <cit.> agrees with what is now believed to be correct to within a factor of 1 + O(1/log T).  Gonek, Hughes, and Keating <cit.> raised the question of whether their model could be extended to predict all the terms of the <cit.> polynomial.

The model of <cit.> has been extended to the function field setting by Bui and Florea <cit.>, and then applied to further families of L-functions by Andrade and  Shamesaldeen <cit.> and Yiasemides <cit.>. Again, in this setting the probabilistic model correctly predicts the leading term of the asymptotic that is conjectured by other methods and known in several cases, but fails to predict the lower-order terms. In this case, the predictions are polynomials in the degree of the conductor (i.e. in N) instead of log T. Theorem <ref> shows that, for q sufficiently large, a probabilistic model based on matrices that are random but not Haar-random improves on this by a power of N.

The idea of integrating against Haar measure times a weight function to calculate the average of a polynomial function on the coefficients function field L-functions appeared earlier in work of Meisner <cit.>, but this work was not probabilistic in nature: the weight function, unlike a probability density function, is not positive (and not even real) and the average must be normalized by a factor depending on the highest weight of the irreducible representations used to express the polynomial (see <ref>).

The idea of restricting the support of Haar measure to obtain more accurate predictions was used in the case of elliptic curve L-functions by Dueñez, Huynh, Keating, Miller, and Snaith <cit.>. Their modification of Haar measure was designed to account for the influence of formulas for the critical value of the L-function that force that value, suitably normalized to be an integer and in particular prevent it from being very close to zero but nonzero. They accordingly considered a measure on random matrices where the critical special value of the characteristic polynomial is prevented from taking small nonzero values. Our adjustment of the probability measure, on the other hand, is designed to account for the influences of small primes, it also involves changing the density and not just the support.

It would be interesting to check the compatibility of our paper with some of these prior works in more detail. First, it should be possible to define an “Euler-Hadamard product" for L-functions in the support of μ_ch. One could then ask how close the distribution of the Euler factors and Hadamard factors is to a product of independent multiplicative function and random matrix distributions at a point (perhaps using an optimal transport distance for probability distributions). If these distributions are close, then not only would μ_ch and <cit.> give similar predictions of the moments, but they would give these predictions for similar reasons.

It would also be enlightening to prove an analogue of <ref> for ratios of L-functions rather than products, using the work of Conrey, Farmer, and Zirnbauer <cit.> to obtain a classical prediction to compare with.

More ambitiously, if an analogue of our construction was made for the family of quadratic Dirichlet characters, and an analogue of <ref> was proven with an error term of size O ( q^ - δ N  ) for δ > 1/4, then one could look for a probabilistic explanation of the secondary terms in moments of quadratic Dirichlet L-functions predicted by multiple Dirichlet series. (If the error term were larger than this, it would dominate the predicted secondary terms, so including them would be meaningless.) It seems unlikely that they could appear for a direct analogue of μ_ch, since these secondary terms ultimately arise from the ability to apply a Poisson summation formula in the modulus of the Dirichlet character and recover a similar sum, and the model of Dirichlet characters based on random multiplicative functions used to construct μ_ch wouldn't reflect this Poisson symmetry, but one could very optimistically hope for a natural modification of μ_ch that predicts these terms. 



 §.§ Motivation, variants, and the number field case


The operation of multiplying the measure of one probability distribution by the density of another, or, equivalently, multiplying the  density of two probability distributions may at first seem strange. However, it has a natural interpretation. Given two different probability distributions μ_1,μ_2 on ℝ^n with continuous probability density functions, we can consider the distribution of a pair of random variables X_1,X_2 independently distributed according to μ_1 and μ_2, and then condition on the event that the distance between μ_1 and μ_2 is at most δ. In the limit as δ→ 0, this conditional distribution will converge to the distribution of two identical random variables, each distributed according to a measure with probability density proportional to the product of the densities of μ_1 and μ_2. 

Thus, multiplying the probability densities arising from random matrices and random Euler products can be seen as, first, generating pairs of random matrices and random Euler products and, second, throwing away those pairs where the characteristic polynomial of the random matrix is not close to the Euler product. This is a plausible way to generate random functions that arise both as characteristic polynomials of matrices and Euler products (as the Dirichlet L-functions L(s,χ) do). However, it cannot be quite right as a model for Dirichlet L-functions, giving the wrong answers in the q→∞ and N→∞ limits. This can be seen most clearly if we let both q and N head to ∞, so the distributions of L_ξ and L_M both converge to the exponentials of random power series with independent complex Gaussian coefficients. Multiplying the probability densities corresponds to squaring the Gaussian probability density function, producing a Gaussian with half the variance. However, to obtain a distribution that interprets between μ_ep and μ_rm, we would like a distribution that converges to the original Gaussian in the q,N →∞ limit. We fix this by dividing by the same Gaussian.

From this heuristic, the right choice of k is not clear. It seems likely that the measure μ_ch does not depend much on the parameter k. The specific value of k chosen makes the analysis as easy as possible, but similar results should be true in a broader range of k.

In fact, if for each value of k we let  F_k(x_1,…, x_k) be the probability density function fo X_1,ξ,…, X_k,ξ and define μ_weighted to be proportional to

    lim_k→∞(  F_k( - q^1/2(M),…, -q^k/2 (M^k)/k)/∏_j=1^k  ( e^  -   (M^j)^2 / j  j / q^j π) ) μ_Haar
 then the same results should be true. This definition is more canonical as it lacks the parameter k, and fits naturally with an infinite-dimensional version of the heuristic for multiplying two probability density functions. However, proving the same results for this measure introduces additional analytical difficulties, starting with proving that the limit as k goes to ∞ exists, that we do not pursue here.

An alternate approach to constructing a measure satisfying <ref> and <ref> is to first check that the pairing ⟨ϕ, ψ⟩ = ∫_ϕψμ_rm is nondegenerate on polynomials in ℂ[c_0,c_1,…, c_0,c_1,…] of degree ≤ k, and using this, verify that there exists a unique ψ∈ℂ[c_0,c_1,…, c_0,c_1,…] of degree ≤ k such that 
    ∫_U(N) ϕψμ_rm  =  ∫_ϕμ_ep
 for all ϕ∈ℂ[c_0,c_1,…, c_0,c_1,…] of degree ≤ k, note that ψ is real-valued, and then consider the signed measure ψμ_rm, for which <ref> and <ref> hold with vanishing error term. The main difficulty with this approach is that the signed measure may not actually be a measure, as the function ψ may be negative on the support of μ_rm. It is easy to check that ψ is nonnegative as long as q is sufficiently large with respect to N, but for q fixed,  ψ is negative even for small values of N. If we view ψμ_rm as an approximation of the true distribution of L(s,χ), the problem is clear: since L(s,χ) is supported on power series with first coefficient c_1 satisfying c_1≤ q, while μ_rm is supported on power series with c_1≤ q^1/2 N, as long as q< N^2, the true measure vanishes on a large region where μ_rm is supported, and so ψ is approximating a function which is zero on that region. Since polynomials cannot be zero on a region without being identically zero, polynomial approximations of functions zero on a region will tend to oscillate between positive and negative values on that region. Thus the signed measure ψμ_rm is rarely a measure. However, this argument suggests that, given that <ref> shows that μ_ch has more reasonable support, it may be possible to multiply μ_ch by a low-degree polynomial to improve the error term in <ref> without compromising positivity.

Whether the strategy of this paper can be applied in the number field context is not yet clear. The fundamental difficulty seems to be that a number field L-function contains much more information than a function field L-function, so it is harder for the supports of distributions arising from random Euler products and random matrix models to intersect.

Let us make this more precise. Consider the problem of defining a random holomorphic function in a variable s whose properties approximate the behavior of ζ(s+it) for t random near a given value T. The basic steps are to define an analogue of the random matrix model, define a random Euler product model, and combine them. Since the function n^it behaves like a random multiplicative function with absolute value 1 and values at the primes independently uniformly distributed on the unit circle, we can again use the Dirichlet series of random completely multiplicative functions as our Euler products. Whatever our random matrix model looks like, the holomorphic functions it produces will probably have functional equations (since the characteristic polynomials of unitary and Hermitian matrices each satisfy a functional equation, and we are trying to approximate the zeta function, which satisfies a functional equation). One natural functional equation to choose is f(1-s) =ϵ   (T/2π)^1/2-s  f(s) since this is consistent with a holomorphic function having zeroes on the critical line distributed with frequency (1/2π) log (T/2π) – in other words, the frequency of zeroes of zeta near T. However, it is easy to see that there are no multiplicative functions whose Dirichlet series satisfy that functional equation, as it forces the coefficients of n^-s to vanish for n> T/2π. So the intersection of the support of the distribution of any random matrix characteristic polynomials having one of these functional equations with the support of the distribution of Dirichlet series of random multiplicative functions will simply vanish, and any attempt to multiply the densities of these distributions will produce a zero distribution.

A similar conclusion can be drawn if we keep the usual functional equation of the Riemann zeta function. In that case, it follows from Hamburger's theorem <cit.> that the intersection of the supports will contain only the actual shifts of the original Riemann zeta function, and so searching for a distribution supported on the intersection, or multiplying the densities, will simply produce the distribution of random shifts of the zeta function. Of course, it is pointless to model these shifts using the shifts themselves.

Observing this problem immediately suggests the rough form of the solution. Rather than looking for a distribution supported on the intersection of the supports of the distributions, we should look for a distribution supported on points which are close to the support of both distributions. In other words, in the heuristic for the product of two probability densities as a δ→ 0 limit, we should avoid taking the limit and instead fix a value of δ. Of course, the nature of this depends on exactly how we define the distance between two holomorphic functions. A natural choice is to integrate the square of the absolute value of the difference between their logarithms against some measure on a subset of the complex plane where they are both defined, but we have a great deal of choice on the measures.

In fact, rather than conditioning the joint distribution of the characteristic polynomial of a random matrix and random Euler product on the event that the two holomorphic functions are close, it seems better to weight the joint probability distribution by the exponential of minus the square of the distance, or another quadratic form in the two functions, before normalizing by a constant to have the total mass one. This weighting sends Gaussians to Gaussians, and should be normalized so that inputting Gaussian approximations to the two distributions outputs a joint distribution whose marginals are the original Gaussians, coupled so that with high probability the two holomorphic functions take similar values at points near 0 to the right of the critical line. (We cannot compare them on the critical line itself since the random Euler products admit a natural boundary there). But it is not clear if there is a single natural coupling to work with. 

In physics, one can consider the eigenvalues of random matrices as being a statistical mechanics model of particles, either on the line or the unit circle, that repel each other and thus have a lesser probability of being close together than independent random points. Specifically, the probability density function should be the exponential of a negative constant times the energy of the system, so the terms in the Weyl integration formula involving the difference of two eigenvalues correspond to a contribution to the energy depending on the distance between two points. We can view this type of exponentially-weighted joint distribution as a statistical mechanics model of points on the critical line together with values ξ(p) on the unit circle for each prime, where, in addition to interacting with each other, the points interact with ξ(p). For the zeta function itself, the interaction is infinitely strong, to the extent that the primes determine the zeroes and the zeroes determine the primes. By choosing an interaction whose strength is not too large and not too small, we may be able to construct a model of the Riemann zeta function whose properties are amenable to computation.

Regardless, this approach produces a joint distribution of two holomorphic functions, one the characteristic polynomial of a matrix and the other an Euler product, but to model the Riemann zeta function we only want one. The simplest approach is to throw out the Euler product, since its natural boundary on the critical line makes it inappropriate for modeling the zeta function on the line, but it may be possible to combine them in a subtler way. 

The exact random matrix model to use is of course a question. A good choice might be to take the characteristic polynomial of a random unitary matrix and plug in e^log T/2π/log N(1/2-s ). This produces a holomorphic function on the whole complex plane with zeroes on the critical line with the correct zero density. Since it is periodic in the imaginary axis, it can't be a good model for the large-scale behavior of the Riemann zeta function, but as long as N is somewhat larger than log T it may be a good model for the local behavior. (The models of <cit.> require setting N very close to log T/2π, but coupling with the Euler product will damp oscillations with frequencies less than that of the leading term 2^-s, allowing us to take larger values of N without getting obviously wrong predictions, and taking larger values of N seems necessary to accurately approximate the contribution of the 2^-s term.) However, we could also consider the eigenvalues of random Hermitian matrices of fixed size, or point processes on the whole critical line. (The determinantal point process associated to the sine kernel, which is the large N limit of random matrices, is not useful for this, as its “characteristic polynomial" is not a well-defined holomorphic function, basically because the distribution of the characteristic polynomial of a random matrix, normalized to keep the frequency of zeroes constant, doesn't converge in the N →∞ limit, but another point process might work.)

One can optimistically hope that there is some reasonably natural way of making the sequence of choices discussed above for which an analogue of <ref> or, ideally, <ref> can be proven. Proving this should be analytically more difficult than <ref>, since the two distributions we are trying to combine are further from each other and further from the Gaussian model and thus showing that the combination has the desired properties of each one should be more difficult, so proving the strongest possible form of <ref> might be a stepping stone to handling the number field case.




 §.§ Geometric and probabilistic approaches to L-functions
 

The probabilistic model μ_ch is compatible with the geometric and representation-theoretic approach to the moments of L-functions suggested by the same author in <cit.>. Specifically, from the geometric perspective the most natural test functions to integrate against are the characters of irreducible representations of U(N), which may be expressed as polynomials in the coefficients of the characteristic polynomial L_M using the fundamental theorem of symmetric polynomials, or, more explicitly, the second Jacobi-Trudi identity for Schur polynomials <cit.>.

Conversely, any polynomial in the coefficients of L_M can be expressed as a linear combination of characters of irreducible representations.  We will check in <ref> that the polynomials of degree ≤ k are exactly the linear combinations of characters whose highest weights, expressed as an N-tuple of integers, have absolute value summing to a number ≤ k. Thus, by orthogonality of characters, irreducible representations whose highest weights have absolute value sums >k are orthogonal to all polynomials of degree ≤ k.

Hence <ref> applies to the characters of irreducible representations with small highest weight, showing that the averages of these functions over μ_ch match their averages over μ_ep, while <ref> applies to the characters of irreducible representations with large highest weight, showing that the averages of these functions over μ_ch cancel. 

By Weil's Riemann hypothesis, every L-function L(s+it,χ) can be expressed as L_M for M∈ U(N) unique up to conjugacy, so we can interpret characters of irreducible representations of U(M) as functions of L(s+it,χ). For irreducible representations of small highest weight, it is not hard to prove that the averages of their characters over L(s+it,χ) match the averages of the same characters over μ_ep. <cit.> showed that the CFKRS predictions for moments of L-functions could be explained by cancellation in the averages of characters of irreducible representations with large highest weight over the family of L-functions, which could in turn be explained by (hypothetical) vanishing of certain cohomology groups whose traces of Frobenius compute this average. <ref> shows that the cancellation of averages of characters of irreducible representations with large highest weight could also be explained by the probabilistic model μ_ch. So this cancellation could have both probabilistic and geometric explanations. (However, note that the amount of cancellation that one can prove in the probabilistic model is different from the amount one can prove under geometric hypotheses – at least currently, it is larger for some representations and smaller for others. Thus it is not possible to say the geometric hypothesis implies the probabilistic model, or vice versa.)


Note that the definition of “small highest weight" used in the two contexts is not identical (the definition here is stricter). This is because the average of the character of an irreducible representation over μ_ep decreases with the highest weight of the representation, at least for representations relevant to calculating moments of fixed degree. Thus, as long as the highest weight is not too small, it is possible for the average against another measure both to cancel and to approximate the average against μ_ep, simply because the average of μ_ep is itself small. So whether we state that these averages cancel or approximate μ_ep is a matter of convenience, and how we sort representations into those two buckets can vary with the context.  The only restriction is that, as the error term in our desired estimates shrinks, fewer representations are flexible in this way.

For the case of L-functions of quadratic Dirichlet characters, analysis analogous to <cit.> was conducted by Bergström, Diaconu, Petersen, and Westerland <cit.>. In the quadratic Dirichlet character setting, the L-function is naturally a characteristic polynomial of a conjugacy class in USp(2N), so one considers characters of irreducible representations of USp(N). They derive the CFKRS predictions, or, equivalently in this setting, the highest-order term of the multiple Dirichlet series predictions, from the assumption of cancellation in averages of characters of irreducible representations of USp(N) of large highest weight. They prove a homological stability result which is a topological enhancement of the fact that averages of irreducible representations of small highest weight over L(s+it,χ) (where χ is now a quadratic Dirichlet character) match the averages of the same characters against a suitable analogue of μ_ep. Since the stable cohomology vanishes in low degrees for representations of large highest weight, the vanishing of cohomology groups whose traces of Frobenius compute the average and hence cancellation in the average follows (for q sufficiently large) from a certain uniform homological stability statement, later proven by Miller, Patzt, Petersen, and Randal-Williams <cit.>.



 §.§ Proof sketch

We now sketch the proofs of the main theorems. Recall that in the definition of μ_weighted we take the Haar measure and multiply by the probability density function of X_1,ξ,…, X_k,ξ divided by a Gaussian probability density function. A key observation is that, if we instead took a suitable Gaussian measure and multiplied by the probability density function of X_1,ξ,…, X_k,ξ divided by a Gaussian probability density function, the density of the Gaussian would cancel and we would obtain the distribution of X_1,ξ,…, X_k,ξ. For this modified measure, the expectation of a low-degree polynomial matches its expectation against μ_ep simply because X_1,ξ,…, X_k,ξ are the coefficients of the random power series log L_ξ distributed according to μ_ep. (The low degree assumption is necessary here because high-degree polynomials may involve coefficients of the power series beyond the first k and thus can't be expressed as functions of X_1,ξ,…, X_k,ξ.)

So proving <ref> is a matter of proving that the expectation of low-degree polynomials is not changed much by the fact that we used the Haar measure instead of the Gaussian measure to construct μ_weighted and μ_ch. It thus crucially requires a bound for the difference, in some sense, between the Haar measure and the Gaussian measure. We rely on the work of Johansson and Lambert <cit.>, who proved a bound for the total variation distance between these distributions. Multiplying a measure by a continuous function can increase the total variation distance proportionally to the sup-norm of the function, so applying this result in our setting requires bounding the sup-norm of the multiplier 
    F( - q^1/2(M),…, -q^k/2 (M^k)/k)/∏_j=1^k  ( e^  -   (M^j)^2 / j  j / q^j π)  .
  This requires pointwise bounds for the probability density function F(x_1,…,x_k) which decrease rapidly as x_1,…, x_k grows.

To obtain pointwise bounds for F(x_1,…,x_k), we first bound the integrals of F(x_1,…,x_k) against a a complex exponential function of x_1,…,x_k. Taking the Fourier transform, i.e. integrating against an imaginary exponential function of x_1,…,x_k would be sufficient if we only wanted a bound for F(x_1,…,x_k) which is uniform in x_1,…, x_k, while integrating against a real exponential function would be sufficient if we wanted a bound for the the integral of F(x_1,…, x_k) over a large region which decreases rapidly as the region becomes more distant from 0, but since we are interested in bounds that are both pointwise and rapidly decreasing we require exponentials of complex-valued functions. The advantage of studying these exponential integrals is that the definition of F as the probability density function of a sum of independent random variables immediately gives a factorization of the exponential integral as a product of simpler integrals, in this case over the unit circle. Thus, a large part of our proof involves proving elementary bounds for these exponential integrals over the unit circle, and then multiplying them together to obtain bounds for integrals of F.

For <ref>, on the other hand, the statement becomes trivial if we replace the multiplier (<ref>) in the definition of μ_weighted and μ_ch by any polynomial of degree ≤ k in the coefficients of a power series. Thus proving <ref> is a matter of finding a suitable approximation of 
    F( x_1,…, x_k )/∏_j=1^k  ( e^  -    |x_j|^2 / j q^j  j / q^j π)
 by a low-degree polynomial in x_1,…, x_k, x_1,…, x_k and bounding the error of this approximation. We choose an approximation in the L^2 sense, with the L^2 norms calculated against the Gaussian measure. (We again use the results of <cit.> to compare the Gaussian measure to the Haar measure). The optimal L^2 approximation against the Gaussian measure can be obtained using the orthogonal polynomials for the Gaussian measure, the Hermite polynomials: Since they form an orthogonal basis, any L^2 function can be written as a linear combination of them, and then one truncates the linear combination by taking only the low-degree polynomial terms, leaving the coefficients of the high-degree polynomials as an error. Bounding the error of this approximation is equivalent to bounding the coefficients of Hermite polynomials of high degree in the Hermite polynomial expansion of (<ref>). These coefficients are naturally expressed as contour integrals of exponential integrals of F(x_1,…,x_k) and we can again bound them by bounding the exponential integrals.

Finally, <ref> is obtained by expressing ∏_i=1^ L(s_i ) ∏_i=+1^ + L ( s_i ) as a polynomial in the coefficients of L and their complex conjugates, breaking that polynomial into low-degree terms and high-degree terms (using irreducible representations, as in <ref>) and applying <ref> to the low-degree terms and <ref> to the high-degree terms.



 §.§ Acknowledgments


The author was supported by NSF grant DMS-2101491 and a Sloan Research Fellowship while working on this project.

I would like to thank Amol Aggorwal, David Farmer, Jon Keating, and Andrei Okounkov for helpful conversations.







§ RANDOM EULER PRODUCTS

The variables, X_1,ξ,…, X_n,ξ are valued in ℂ, but it will be convenient for us to treat them as valued in ℝ^2 by viewing complex numbers as real vectors in the usual way, taking their real and imaginary parts as coordinates. This is because we will mainly be interested in their dot products with other vectors in ℝ^2, which can be expressed in terms of complex numbers but less directly.

To that end, we give a formula for X_n,ξ as a vector in ℝ^2.  For each prime polynomial 𝔭, let θ_𝔭 be the argument of ξ(𝔭), so that (<ref>) gives

    X_n,ξ = ∑_d | n∑_𝔭∈𝔽_q[u]^+ 
    𝔭 =d  
    primed/ne^ i n/dθ_𝔭 .


Now for θ∈ℝ let  
    (θ) = [ cosθ; sinθ ]
 be e^ iθ∈ℂ viewed as a vector in ℝ^2, so that we have

    X_n,ξ = ∑_d | n∑_𝔭∈𝔽_q[u]^+ 
    𝔭 =d  
    primed/n( n/dθ_𝔭 ) .



 Our first goal will be to upper bound F(x_1,…, x_k). Two basic tools to do this are the Fourier transform of F(x_1,…,x_k), i.e. the characteristic function of X_1,ξ,…, X_k,ξ, represented by the expectation

    𝔼 [ e^  i ∑_n=1^k X_n,ξ· w_n  ]

for vectors w_1,…, w_n ∈ℝ^2, and the Laplace transform of F(x_1,…,x_k), i.e. the moment generating function of X_1,ξ,…, X_k,ξ, represented by the expectation

    𝔼 [ e^∑_n=1^k X_n,ξ· v_n  ]

for vectors v_1,…, v_n ∈ℝ^2. We will in fact need a hybrid of these, also referred to as the Laplace transform, expressed as 

    𝔼 [ e^∑_n=1^k X_n,ξ· v_n  +  i ∑_n=1^k X_n,ξ· w_n  ].

We could equivalently express  X_n,ξ· v_n  +  i ( X_n,ξ· w_n) as  z_n,1   X_n,ξ + z_n,2X_n,ξ for a certain pair of complex numbers z_n,1, z_n,2 depending real-linearly on v_n and w_n, but this would be unwieldy for the calculations we want to do, which focus on the size of these expectations, as we want to separate out the parameters v_n which affect the size of the exponential from the parameters w_n which affect only its argument. Thus it is better for our purposes to work with dot products in ℝ^2.

Let E_d be the number of prime polynomials of degree d in 𝔽_q[u]^+.


 Let v_1,…, v_k and w_1,…, w_k be vectors in ℝ^2. Then

    𝔼 [ e^∑_n=1^k X_n,ξ· v_n  +  i ∑_n=1^k X_n,ξ· w_n  ] = ∏_d=1^k (  ∫_0^2π   e^∑_m=1^⌊k/d⌋ ((m θ)  · v_md + i ( (m θ)· w_md ))/m  dθ/2π)^E_d .
 

 From (<ref>) we have

    ∑_n=1^k X_n,ξ· v_n= ∑_n=1^k ∑_d | n∑_𝔭∈𝔽_q[u]^+ 
    𝔭 =d  
    primed/n( n/dθ_𝔭 ) · v_n

which writing m= n/d and switching the order of summation is 

    ∑_d=1^k   ∑_𝔭∈𝔽_q[u]^+ 
    𝔭 =d  
    prime∑_m=1^⌊k/d⌋( m θ_𝔭 ) · v_md/m

so

    e^∑_n=1^k X_n,ξ· v_n    = ∏_d=1^k ∏_𝔭∈𝔽_q[u]^+ 
    𝔭 =d  
    prime e^∑_m=1^⌊k/d⌋(m θ_𝔭)  · v_md /m.

An analogous identity holds with w_n. Since θ_𝔭 are independent for different 𝔭 and uniformly distributed in [0,2π], we have

    𝔼 [ e^∑_n=1^k X_n,ξ· v_n  +  i ∑_n=1^k X_n,ξ· w_n  ] =𝔼[ ∏_d=1^k ∏_𝔭∈𝔽_q[u]^+ 
    𝔭 =d  
    prime e^∑_m=1^⌊k/d⌋ ((m θ_𝔭)  · v_md + i ( (m θ_𝔭)· w_md ))/m  ]


    = ∏_d=1^k ∏_𝔭∈𝔽_q[u]^+ 
    𝔭 =d  
    prime𝔼[ e^∑_m=1^⌊k/d⌋  ((m θ_𝔭)  · v_md + i ( (m θ_𝔭) · w_md ))/m  ]


    =   ∏_d=1^k  ∏_𝔭∈𝔽_q[u]^+ 
    𝔭 =d  
    prime(  ∫_0^2π   e^∑_m=1^⌊k/d⌋  ((m θ)  · v_md + i ( (m θ)· w_md ))/m dθ/2𝔭) .
  

In view of <ref>, we will begin by estimating ∫_0^2π   e^∑_m=1^⌊k/d⌋ ((m θ)  · v_md + i ( (m θ)· w_md ))/mdθ/2π, starting with the case where w_n=0 for all n before handling the general case. This will require different techniques to provide useful estimates with v_1,w_1 in different ranges.



 §.§ Real exponential integrals


This subsection is devoted to estimating ∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m dθ/2π. We have expanded the finite sum to an infinite sum because our estimates need to be uniform in the length of the sum and bounds uniform in the length of the sum are equivalent to bounds in the infinite sum case but the infinite sum statements are slightly more elegant and general.  A simple guess for the average of this sum, based on a second-order Taylor expansion, is e^∑_m=1^∞v_m^2/4m^2. Our goal will be to prove a bound of roughly this shape, though our final bound will be worse in some ranges and better in others. 

Our rough strategy to estimate ∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m dθ/2π is to use an argument controlling the error of a Taylor series when the variables are small and the trivial bound  e^∑_m=1^∞(m θ) · v_m/m ≤  e^∑_m=1^∞v_m/m when the variables are large. The argument needs to be more complex since we have infinitely many variables, some of which may be small and sum of which may be large. We first handle the case where v_m=0 for all m>1  in Lemma <ref>, where we obtain a savings over the simple guess e^v_1^2/4 using a finite Taylor series in the small range, the trivial bound in the large range, and a different power series argument in an intermediate range. This savings will be very convenient throughout the argument as by shrinking it slightly we can absorb unwanted terms from other estimates. In <ref> we make a more complicated, multivariable Taylor series estimate. This is expressed in terms of a ratio of integrals to allow us to preserve the savings. Finally in <ref> we combine these estimates and use a version of the trivial bound that allows us to ignore an individual v_m if it is too large. 

 There exists δ_1>0 such that for all v_1∈ℝ^2 we have

    log∫_0^π e^(θ) · v_1dθ/2π≤v_1^2/4 - δ_1 min ( v_1^4, v_1^2) .



 It is equivalent to show that the function 
    v_1^2/4 - log∫_0^2π e^(θ) · v_1d θ/2π/min ( v_1^4, v_1^2)
 has a lower bound δ_1>0 for all v_1∈ℝ^2∖{0}.

We first check that (<ref>) is positive on all of ℝ^2 ∖{0}. To do this, we use the power series

    log∫_0^2π e^(θ) · v_1d θ/2π  = log∑_d=0^∞v_1^2d/ (d!)^2 2^2d .

and note that (<ref>) is strictly less than  
    v_1^2/4  = log e^v_1^2/4 = log∑_d=0^∞v_1^2d/ (d!) 2^2d.
 It follows immediately that (<ref>) is positive.

Next, using the first couple terms of the Taylor series for logarithm, we compute the Taylor series of  (<ref>) as v_1^2/4 - v_1^4/64 + … and conclude that (<ref>) is equal to v_1^2/4  - v_1^4/64 + O ( v_1^6/6) for v_1 small. Plugging this into (<ref>), we see that (<ref>) converges to 1/64 as v_1 goes to ∞.

Finally, e^(θ) · v_1≤ e^v_1 so that log∫_0^2π e^(θ) · v_1d θ/2π≤v_1 and thus for v_1≥ 1, (<ref>) is at least 
    v_1^2/4 -   v_1/v_1^2 = 1/4 - 1/v_1
 and thus converges to 1/4 as v_1 goes to ∞.

Thus (<ref>), a continuous function on ℝ^2 ∖{0}, is positive everywhere and bounded away from 0 in both a neighborhood of 0 and a neighborhood of ∞. By compactness of ℝ^2 ∪{∞}, it follows that (<ref>) has a lower bound δ_1>0. 

To apply Taylor's theorem in the case of infinitely many variables, we will need some trick to relate the power series of a function in many variables to the power series of a function in fewer variables. This may be accomplished using the following lemma.
 Let g_1 and g_2 be power series in one or more variables, with constant coefficients 1, such that g_1 has nonnegative coefficients and the coefficient of each monomial in g_1 is greater than or equal to the coefficient of the corresponding monomial in g_2.  Then -log(2-g_1) has nonnegative coefficients and the coefficient of each monomial in -log(2-g_1) is greater than or equal to the coefficient of the corresponding monomial in log g_2. 

 We have 
    log (g_2) = log (1 + (g_2-1)) =  (g_2-1) -  (g_2-1)^2/2 +  (g_2-1)^3/3 -   (g_2-1)^4/4 + ….
 
Writing each term in g_2-1 as a sum of coefficients times monomials, and bounding each coefficient by the corresponding coefficient in g_1-1, we see that the coefficient of any monomial in this expression is at most the coefficient of the same monomial in

    (g_1-1) +  (g_1-1)^2/2 +  (g_1-1)^3/3 +  (g_1-1)^4/4 + … = - log ( 1- (g_1-1)) =-log(2-g_1).
 


 Let × denote multiplication of complex numbers viewed as a multiplication operation for vectors in ℝ^2.

 Let (v_m)_m=1^∞ be a sequence of vectors in ℝ^2. For any n let u_n = ∑_m=n^∞v_m/m.  Then as long as u_1< 1/2 we have

    log∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m dθ/2π  - log∫_0^π e^(θ) · v_1dθ/2π  = ∑_m=2^∞v_m^2/4m^2 +   (v_1 × v_1) · v_2/16 + O( u_1 u_2 u_3 + u_1^3 u_2) .



 We use [x_1^n_1 x_2^n_2] to denote extracting the coefficient of x_1^n_1 x_2^n_2 in a power series in x_1,x_2. We begin with the observation

    e^ x_1 (θ) · v_1   + x_2 ∑_m=2^∞(m θ) · v_m/m [x_1^n_1 x_2^n_2] ≤v_1^n_1u_2^n_2 / (n_1! n_2!) = e^v_1 x_1+ u_2 x_2 [x_1^n_1 x_2^n_2]

which implies by linearity

    ∫_0^2π   e^ x_1 (θ) · v_1   + x_2 ∑_m=2^∞(m θ) · v_m/m dθ/2π [x_1^n_1 x_2^n_2] ≤  e^v_1 x_1+ u_2 x_2 [x_1^n_1 x_2^n_2] .


From Lemma <ref> we obtain

    log∫_0^2π   e^ x_1 (θ) · v_1   + x_2 ∑_m=2^∞(m θ) · v_m/m dθ/2π [x_1^n_1 x_2^n_2] 
    ≤     - log (2 - e^v_1 x_1+ u_2 x_2 ) [x_1^n_1 x_2^n_2].


We have (evaluating a power series at x_1=1,x_2=1)

    log∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m dθ/2π   = ∑_n_1=0^∞∑_n_2=0^∞log∫_0^2π   e^ x_1 (θ) · v_1   + x_2 ∑_m=2^∞(m θ) · v_m/m dθ/2π [x_1^n_1 x_2^n_2]

and (evaluating a power series at x_1=1,x_2=0)

    log∫_0^π e^(θ) · v_1dθ/2π =  ∑_n_1=0^∞log∫_0^2π   e^ x_1 (θ) · v_1   + x_2 ∑_m=2^∞(m θ) · v_m/m dθ/2π [x_1^n_1 x_2^0]

so that

    log∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m dθ/2π  - log∫_0^π e^(θ) · v_1dθ/2π
      = ∑_n_1=0^∞∑_n_2=1^∞   log∫_0^2π   e^ x_1 (θ) · v_1   + x_2 ∑_m=2^∞(m θ) · v_m/m dθ/2π [x_1^n_1 x_2^n_2] .




We split the sum in (<ref>) up into terms with n_1 + n_2 ≤ 3, which we evaluate, and the terms with n_1 + n_2 ≥ 4, which we bound.

For any b_1 ∈ [0, v_1], we observe that (b_1,u_2) lies in the compact set { (a_1,a_2)∈ℝ^2 | a_1≥ 0, a_2 ≥ 0, a_1+a_2 ≤1/2} where the function ∂/∂ a_2log ( 2- e^ a_1 +a_2) is smooth. Thus by Taylor's theorem we have

    ∑_n_1, n_2 ≥ 0 
     n_1+n_2 ≥ 3∂/∂ a_2log (2 - e^a_1 + a_2  ) [a_1^n_1 a_2^n_2]  v_1^n_1 b_2^n_2 = O( ( v_1 +b_2 )^3) = O( (v_1 + u_2)^3) = O(u_1^3)
 since the left-hand side of (<ref>) is the error in the second-order Taylor approximation to the function ∂/∂ a_2log (2 - e^a_1 + a_2  ) at the point v_1, b_2 and the constant in Taylor's theorem is uniform by compactness.

By (<ref>) and (<ref>) we have 
 
    ∑_n_1≥ 0, n_2 > 0 
     n_1+n_2 ≥ 4log∫_0^2π   e^ x_1 (θ) · v_1   + x_2 ∑_m=2^∞(m θ) · v_m/m dθ/2π [x_1^n_1 x_2^n_2]

 
    ≤∑_n_1≥ 0, n_2 > 0
     n_1+n_2 ≥ 4log∫_0^2π   e^ x_1 (θ) · v_1   + x_2 ∑_m=2^∞(m θ) · v_m/m dθ/2π [x_1^n_1 x_2^n_2]

 
    ≤ -  ∑_n_1≥ 0, n_2 > 0 
     n_1+n_2 ≥ 4log (2 - e^v_1 x_1+ u_2 x_2 ) [x_1^n_1 x_2^n_2]=  -  ∑_n_1≥ 0, n_2 > 0 
     n_1+n_2 ≥ 4log (2 - e^a_1 + a_2  ) [a_1^n_1 a_2^n_2]  v_1^n_1 u_2^n_2


    = - ∑_n_1≥ 0, n_2 > 0
     n_1+n_2 ≥ 4log (2 - e^a_1 + a_2  ) [a_1^n_1 a_2^n_2]  ∫_0^u_2  n_2 v_1^n_1 b_2^n_2-1 db_2
 
    = - ∫_0^u_2 n_2  ∑_n_1≥ 0, n_2 > 0 
     n_1+n_2 ≥ 4log (2 - e^a_1 + a_2  ) [a_1^n_1 a_2^n_2]  v_1^n_1 b_2^n_2-1 db_2


    = - ∫_0^u_2∑_n_1, n_2 ≥ 0 
     n_1+n_2 ≥ 3∂/∂ a_2log (2 - e^a_1 + a_2  ) [a_1^n_1 a_2^n_2] v_1^n_1 b_2^n_2 db_2  = ∫_0^u_2 O( u_1^3)  db_1 = O( u_1^3 u_2) .



We evaluate the terms with n_1 + n_2 ≤ 3 by Taylor expanding each term and integrating to obtain

    ∫_0^2π   e^ x_1 (θ) · v_1   + x_2 ∑_m=2^∞(m θ) · v_m/m dθ/2π
 
    = 1 +   v_1^2/4 x_1^2 + ∑_m=2^∞v_m^2/4m^2 x_2^2 +   (v_1 × v_1) · v_2/16 x_1^2 x_2 +  ∑_m=2^∞ (v_1 × v_m) · v_m+1/ 8 m (m+1) x_1 x_2^2+ ∑_m_1,m_2=2^∞ (v_m_1× v_m_2) · v_m_1+m_2/ 16 m_1 m_2(m_1+m_2) x_2^3 + …

Taking logarithms of both sides, we obtain

    log∫_0^2π   e^ x_1 (θ) · v_1   + x_2 ∑_m=2^∞(m θ) · v_m/m dθ/2π
 
    =   v_1^2/4 x_1^2 + ∑_m=2^∞v_m^2/4m^2 x_2^2 +   (v_1 × v_1) · v_2/16 x_1^2 x_2 +  ∑_m=2^∞ (v_1 × v_m) · v_m+1/ 8 m (m+1)x_1 x_2^2 + ∑_m_1,m_2=2^∞ (v_m_1× v_m_2) · v_m_1+m_2/ 16 m_1 m_2(m_1+m_2) x_2^3 + …

and ignoring the terms with exponent of x_2 zero and then substituting x_1 ,x_2=1, we obtain 

    ∑_n_1≥ 0, n_2 > 0 
     n_1+n_2 ≤ 3log∫_0^2π   e^ x_1 (θ) · v_1   + x_2 ∑_m=2^∞(m θ) · v_m/m dθ/2π [x_1^n_1 x_2^n_2]

    = ∑_m=2^∞v_m^2/4m^2 +   (v_1 × v_1) · v_2/16 +  ∑_m=2^∞ (v_1 × v_m) · v_m+1/ 4 m (m+1) + ∑_m_1,m_2=2^∞ (v_m_1× v_m_2) · v_m_1+m_2/ 8 m_1 m_2(m_1+m_2).

This gives (<ref>) once we check that

    ∑_m=2^∞ (v_1 × v_m) · v_m+1/ 4 m (m+1) + ∑_m_1,m_2=2^∞ (v_m_1× v_m_2) · v_m_1+m_2/ 8 m_1 m_2(m_1+m_2)  = O( u_1 u_2 u_3)

which is clear since


    ∑_m=2^∞ (v_1 × v_m) · v_m+1/ 4 m (m+1) + ∑_m_1,m_2=2^∞ (v_m_1× v_m_2) · v_m_1+m_2/ 8 m_1 m_2(m_1+m_2)


    ≤∑_m=2^∞v_1v_mv_m+1/ 4 m (m+1) + ∑_m_1,m_2=2^∞v_m_1v_m_2v_m_1+m_2/ 8 m_1 m_2(m_1+m_2)


    ≤∑_m_1=1^∞∑_m_2=2^∞v_m_1v_m_2v_m_1+m_2/ 4 m_1 m_2(m_1+m_2)


    ≤∑_m_1=1^∞∑_m_2=2^∞∑_m_3=3^∞v_m_1v_m_2v_m_3/4m_1m_2m_3 =  u_1 u_2 u_3/4 .




  There exists δ_2>0 and C_1≥ 1 such that, for (v_m)_m=1^∞ a sequence of vectors in ℝ^2 with ∑_m=1^∞v_m^2<∞, we have

    log∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m dθ/2π≤v_1^2/4 - δ_2 min ( v_1^4, v_1^2) + ∑_m=2^∞min ( C_1 v_m^2, v_m/m ).
 

 A key fact we will use multiple times is that replacing v_m by 0 decreases the left hand side of (<ref>) by at most v_m/m, since it shrinks the integrand e^∑_m=1^∞(m θ) · v_m/m at each point by a factor of at most e^v_m/m and thus, because the integrand is positive, shrinks the integral by a factor of at most e^v_m/m.

For m≥ 2, if v_m≥1/mC_1 then min ( C_1 v_m^2, v_m/m ) = v_m/m. If we replace v_m by 0, the left side of (<ref>) decreases by at most v_m/m while the right side decreases by exactly v_m/m so the bound after making the change implies the bound before. Repeating this for all m, we may assume 
    v_m< 1/m C_1
 for all m≥ 2.

Take δ_1 as in <ref> and then set  δ_2 =δ_1/2 so that by <ref> we have 
    log∫_0^π e^(θ) · v_1dθ/2π≤v_1^2/4 - 2δ_2 min ( v_1^4, v_1^2)
 so if we let 

    Disc (v_1,v_2,…) =  log∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m dθ/2π - log∫_0^π e^(θ) · v_1dθ/2π

then it suffices to check for C_1 sufficiently large that

    Disc (v_1,v_2,…)    ≤δ_2 min ( v_1^4, v_1^2) +   ∑_m=2^∞min ( C_1 v_m^2, v_m/m ).

We  have 
    Disc (v_1,v_2,…)  ≤∑_m=2^∞v_m/m ≤1/C_1 (ζ(2)-1)
 by the key fact and (<ref>). Thus we may assume that

    δ_2 min ( v_1^4, v_1^2)  ≤1/C_1 (ζ(2)-1).
 because otherwise (<ref>) holds automatically. Combining (<ref>) and (<ref>) gives 

    ∑_m=1^∞v_m/m = v_1 + ∑_m=2^∞v_m/m≤max( (  δ_2^-11/C_1 (ζ(2)-1))^1/4, (  δ_2^-11/C (ζ(2)-1))^1/4) + 1/C_1 (ζ(2)-1)
 and choosing C_1 sufficiently large, the right hand side of (<ref>) is <1/2, and thus we may apply <ref>, obtaining

    Disc (v_1,v_2,…) =  ∑_m=2^∞v_m^2/4m^2 +   (v_1 × v_1) · v_2/16 + O( u_1 u_2 u_3 + u_1^3 u_2).

We now simplify (<ref>) by bounding the terms appearing on the right hand side. To do this, we we use the facts clear from the definitions that u_1 = v_1 + u_2 and u_3≤ u_2 as well as the assumptions (<ref>) and (<ref>) that imply u_2 and v_1, respectively, are bounded by constants. These facts imply

    (v_1 × v_1) · v_2/16≪v_1^2 v_2≤v_1^2 u_2


    O(u_1u_2 u_3) ≪ u_1 u_2 u_3 ≤ u_1 u_2^2 = (v_1+ u_2) u_2^2 ≪ u_2^2


    O(u_1^3 u_2) ≪ u_1^3 u_2 = (v_1 + u_2)^3 u_2 = v_1^3 u_2 +3 v_1^2 u_2^2 + 3 v_1u_2^3 + u_2^4 ≪v_1^2 u_2 + u_2^2

giving

    Disc (v_1,v_2,…) =  ∑_m=2^∞v_m^2/4m^2 +  O (  v_1^2 u_2 + u_2^2) .

Applying the completing-the-square-bound v_1^2 u_2 ≤ϵv_1^4 + 1/4ϵ u_2^2 for some sufficiently small ϵ, we obtain

    Disc (v_1,v_2,…) ≤∑_m=2^∞v_m^2/4m^2    + δ_2 v_1^4+ O(u_2^2) .

Applying the Cauchy-Schwarz bound u_2^2 ≤ (ζ(2)-1) ∑_m=2^∞v_m^2 we obtain

    Disc (v_1,v_2,…) ≤δ_2 v_1^4 + O (  ∑_m=2^∞v_m^2)  .

Finally (<ref>) implies (<ref>) because (<ref>) gives v_1≤ 1 (for C_1 sufficiently large) so δ_2 v_1^4 = δ_2 min ( v_1^4, v_1^2) and (<ref>) gives ∑_m=2^∞min ( C_1v_m^2, v_m/m )= ∑_m=2^∞ C_1 v_m^2 which dominates any expression of the form O (  ∑_m=2^∞v_m^2) as long as C_1 is sufficiently large. 








 §.§ Complex exponential integrals


This subsection is devoted to bounding the integral  ∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m + i∑_m=1^∞(m θ) · w_m/m.  We have three different estimates that roughly handle three different ranges for w_1. When w_1 is small we will apply Lemma <ref> which is proven using a Taylor series argument. When w_1 is large we will apply Lemma <ref> which is proven using a stationary phase argument. When w_1 is intermediate we will apply Lemma <ref> which is proven using a more elementary argument involving the range of values attained by the function ∑_m=1^∞(m θ) · w_m/m. 

We then multiply the bounds together to obtain bounds for the expectation 𝔼 [ e^∑_n=1^k X_n,ξ· v_n  +  i ∑_n=1^k X_n,ξ· w_n  ], with the final bounds relevant to the remainder of the argument contained in Corollary <ref>.



 Fix δ_3< 1/64. Let (v_m)_m=1^∞ and (w_m)_m=1^∞ be sequences of vectors in ℝ^2. If v_1 + w_1 + ∑_m=2^∞√(v_m^2 + w_m^2)/m <  1/2 then 

    log∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m + i∑_m=1^∞(m θ) · w_m/m dθ/2π
    ≤   v_1^2/4 - w_1^2/4 - δ_3 v_1^4 + O_δ_3  ( v_1^6 + w_1^3 + ∑_m=2^∞ (v_m^2+ w_m^2 )).



  Let W =  ∑_m=2^∞√(v_m^2 + w_m^2)/m. For any θ,  

    e^λ(m θ) · v_1 + iλ^2 (θ) · w_1+ λ^3 ∑_m=1^∞(m θ) · v_m/m + iλ^3 ∑_m=1^∞(m θ) · w_m/m
  is a power series in λ whose coefficient of λ^n is bounded by the coefficient of λ^n in  
    e^λv_1 + λ^2 w_1 + λ^3 W.


Hence the coefficient of λ^n in 
    ∫_0^2π e^λ(m θ) · v_1 + iλ^2 (θ) · w_1+ λ^3 ∑_m=1^∞(m θ) · v_m/m + iλ^3 ∑_m=1^∞(m θ) · w_m/m dθ/2π
 is also bounded by the coefficient of λ^n in (<ref>).

By Lemma <ref>, the coefficient of λ^n in 
    log e^λ(m θ) · v_1 + iλ^2 (θ) · w_1+ λ^3 ∑_m=1^∞(m θ) · v_m/m + iλ^3 ∑_m=1^∞(m θ) · w_m/m dθ/2π
  is bounded by the coefficient of λ^n in the power series  
    - log (2 - e^λv_1+ λ^2 w_1 + λ^3 W).
  We now observe that Taylor's theorem applied to the function - log ( 2 -e^ x +y^2+z^3 ) implies that the sum of all terms of degree ≥ 6 appearing in the power series for - log ( 2 -e^ x +y^2+z^3 ) is O ( x^6+y^6+z^6) uniformly for x,y,z≥ 0 such that x+y^2+z^3 ≤ 1/2. Indeed such x,y,z lie in a compact region where the function is smooth so all derivatives are bounded. Plugging in x=λv_1, y =  λw_1^1/2, z =  λ W^1/3 and then setting λ=1, we obtain the sum of the coefficients of λ^n in (<ref>) for n from 6 to ∞. Hence the sum of the coefficient of λ^n in (<ref>) for n from 6 to ∞ is 
    O (  v_1^6 + w_1^3 + W^2) = O ( v_1^6 + w_1^3 + ∑_m=2^∞ (v_m^2+ w_m^2 )).
  

On the other hand, the coefficients of λ, λ^2, λ^3,λ^4, λ^5 in (<ref>) are, respectively,

    0


    v_1^2/4


    i  v_1 · w_1/2


    v_1^4/64  - w_1^2/4


    (v_1 × v_1) · v_2/ 16+ i  (v_1 × v_1) · w_2/16 + i v_1^2 v_1 · w_1/16 .

Taking logarithms, the coefficients of λ, λ^2, λ^3, λ^4, λ^5 in (<ref>) are identical except the coefficient of λ^4 is

    - v_1^4/64  - w_1^2/4 .


Plugging λ=1 into (<ref>) and taking the real part, this gives

    log∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m + i∑_m=1^∞(m θ) · w_m/m dθ/2π
 
    = v_1^2/4 - w_1^2/4 - v_1^4/64 +  (v_1 × v_1) · v_2/16  +O (v_1^6 + w_1^3 + ∑_m=2^∞ (v_m^2+ w_m^2 )).

which is exactly the desired bound (<ref>) except for the term (v_1 × v_1) · v_2/16, which can be controlled by observing that

    (v_1 × v_1) · v_2/16≤v_1^2 v_2/16≤( 1/64-δ_3) v_1^4 +  1/1024/1/64 - δ_3 v_2^2=( 1/64-δ_3) v_1^4 + O_δ_3( v_2^2) .
  

   Let (v_m)_m=1^∞ and (w_m)_m=1^∞ be sequences of vectors in ℝ^2. If ∑_m=1^∞v_m/m<∞ and ∑_m=1^∞w_m^2 < ∞ then

    ∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m + i∑_m=1^∞(m θ) · w_m/m dθ/2π/∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m dθ/2π
 
    ≤ 1- e^ -2 ∑_m=1^∞v_m/mmin(  ∑_m=1^∞w_m^2/π^2 m^2, 1/ 2∑_m=1^∞w_m^2).
 

 Let ϕ be the argument of ∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m + i∑_m=1^∞(m θ) · w_m/m dθ/2π. We have

    ∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m + i∑_m=1^∞(m θ) · w_m/m dθ/2π
 
    = ∫_0^2π e^ -i ϕ + ∑_m=1^∞(m θ) · v_m/m + i∑_m=1^∞(m θ) · w_m/m dθ/2π


    = ∫_0^2π e^∑_m=1^∞(mθ) · v_m/mcos( -ϕ + ∑_m=1^∞(mθ) · w_m /m ) .


Using the bound 
    e^∑_m=1^∞(m θ) · v_m/m∈ [e^ - ∑_m=1^∞v_m/ m, e^∑_m=1^∞v_m /m  ]
 valid for all θ, we obtain in particular

    ∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m dθ/2π≤  e^∑_m=1^∞v_m /m

so that it suffices to prove

    ∫_0^2π e^∑_m=1^∞(mθ) · v_m/mcos ( -ϕ + ∑_m=1^∞(mθ) · w_m /m ) 
       ≤∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m dθ/2π -  e^ - ∑_m=1^∞v_m/mmin(  ∑_m=1^∞w_m^2/π^2 m^2, 1/ 2∑_m=1^∞w_m^2)




We split into two cases depending on whether e^i ∑_m=1^∞(m θ) · w_m/m = - e^i ϕ for some θ or not.

First, suppose e^i ∑_m=1^∞(m θ_0) · w_m/m = - e^i ϕ for some θ_0. Let x =∑_m=1^∞(m θ_0) · w_m/m and let I be the longest interval around θ_0 on which 
    ∑_m=1^∞(m θ) · w_m/m ∈ [x-π/2, x+π/2].
 Then on the boundary of I, we have ∑_m=1^∞(m θ) · w_m/m = x ±π/2 while at θ_0 ∈ I it takes the value x so

    π≤∫_I  d/dθ∑_m=1^∞(m θ) · w_m/m  d θ = ∫_I  ∑_m=1^∞(m θ+ π/2) · w_m  d θ
 
    ≤√(I∫_0^2π∑_m=1^∞(m θ+ π/2)  · w_m ^2 d θ) = √(Iπ∑_m=1^∞w_m^2)

 so I≥π/∑_m=1^∞w_m^2 but for each θ∈ I we have 
    cos ( -ϕ + ∑_m=1^∞(mθ) · w_m /m )≤ 0
 so
 
    ∫_0^2π e^∑_m=1^∞(mθ) · v_m/mcos ( -ϕ + ∑_m=1^∞(mθ) · w_m /m )

 
    ≤∫_0^2π e^∑_m=1^∞(m θ) · v_m/m dθ/2π - ∫_Ie^∑_m=1^∞(m θ) · v_m/m dθ/2π

 
    ≤∫_0^2π e^∑_m=1^∞(m θ) · v_m/m dθ/2π - I/2π e^ - ∑_m=1^∞v_m/m

 
    ≤∫_0^2π e^∑_m=1^∞(m θ) · v_m/m dθ/2π - 1 /2 ∑_m=1^∞w_m^2  e^ - ∑_m=1^∞v_m/m

giving (<ref>).
 
 Next suppose that e^∑_m=1^∞(m θ) · w_m/m≠  - e^i ϕ for any θ, or in other words ∑_m=1^∞(m θ) · w_m/m ≠ϕ + π n for any odd integer n. After shifting ϕ by an even integer multiple of 2π, we may assume ∑_m=1^∞(m θ) · w_m/m ∈ (ϕ-π, ϕ+π) for all θ. The simple trigonometric inequality cosθ≤   1 - 2 θ^2/π^2 for θ∈ (-π,π) gives
  
    ∫_0^2π e^∑_m=1^∞(mθ) · v_m/mcos ( -ϕ + ∑_m=1^∞(mθ) · w_m /m )

 
    ≤∫_0^2 π e^∑_m=1^∞(m θ) · v_m/m(1 - 2/π^2( ϕ - ∑_m=1^∞(m θ) · w_m/m )^2 )  d θ/2π

  
    = ∫_0^2π e^∑_m=1^∞(m θ) · v_m/mdθ/2π  - 2/π^2∫_0^2π  e^∑_m=1^∞(m θ) · v_m/m( ϕ - ∑_m=1^∞(m θ) · w_m/m )^2 dθ/2π

  
    ≤∫_0^2π e^∑_m=1^∞(m θ) · v_m/mdθ/2π - e^ -∑_m=1^∞v_m/m∫_0^2π( ϕ - ∑_m=1^∞(m θ) · w_m/m )^2 dθ/2π

 
    = ∫_0^2π e^∑_m=1^∞(m θ) · v_m/mdθ/2π - e^ -∑_m=1^∞v_m/m1/π^2( ∑_m=1^∞w_m^2/m^2 + ϕ^2)
 
    ≤∫_0^2π e^∑_m=1^∞(m θ) · v_m/mdθ/2π - e^ -∑_m=1^∞v_m/m1/π^2∑_m=1^∞w_m^2/m^2
  
 giving (<ref>). 




  Let (v_m)_m=1^∞ and (w_m)_m=1^∞ be sequences of vectors in ℝ^2. If ∑_m=1^∞v_m^2 + ∑_m=1^∞w_m^2 < ∞ then

    ∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m + i∑_m=1^∞(m θ) · w_m/m dθ/2π/sup_θ∈ [0,2π]   e^∑_m=1^∞(m θ) · v_m/m 
    ≤   1/√(w_1)( 4/π + 1 +   √(∑_m=1^∞v_m^2 +  ∑_m=2^∞w_m^2)/√(π)w_1^1/4).


 
 
  By shifting θ we may assume w_1 = (w_1,0).  Let 

    ℱ( θ) = e^∑_m=1^∞(m θ) · v_m/m + i∑_m=2^∞(m θ) · w_m/m
 so that

    e^∑_m=1^∞(m θ) · v_m/m + i∑_m=1^∞(m θ) · w_m/m   = e^ i w_1cosθℱ(θ)
  and thus the integral to bound in (<ref>) is ∫_0^2π    e^ i w_1cosθℱ(θ)  dθ/2π. 

We now informally explain the strategy of proof: handle this integral by the method of stationary phase. Ignoring ℱ, one standard form of the stationary phase method is to change variables from θ to cosθ, apply integration by parts, and then reverse the change of variables, giving an integral where the derivative w_1sinθ appears in the denominator. Before doing this, we remove from the integral and handle separately the region where sinθ is so small that this gives a worse bound. Since our desired bound (<ref>) shrinks as w_1 grows but grows in the other variables, we should think of w_1 as large and the other variables as small. In other words, we think of ℱ(θ) as varying more slowly than e^ i w_1cosθ. Because of this, we do not need to modify our change of variables strategy to account for F (which would give a better bound but with a considerably more complicated formula accounting for multiple potential critical points of ∑_m=1^∞(mθ) · w_m/m). 
 
 
 
 
 We first handle the integral from 0 to π.  For each δ with 0< δ < π/2 we have

    | ∫_0^π    e^ i w_1cosθℱ(θ) dθ/2π  -  ∫_δ^π-δ  e^ i w_1cosθℱ(θ) dθ/2π| ≤2 δ/2πℱ_∞

 and the change of variables c = cosθ followed by integration by parts gives
 
    ∫_δ^π-δ   e^ i w_1cosθℱ(θ) dθ/2π
    
     =  ∫_- cosδ^cosδ   e^ i w_1 c ℱ(arccos c )   1/ 2 π√(1-c^2)  dc 
    
      =  - ∫_- cosδ^cosδ    e^ i  w_1c/ i w_1d/dc ( ℱ(arccos c)    1/ 2 π√(1-c^2))  dc  +  e^ i w_1c/ i w_1ℱ(arccos c)   1/ 2 π√(1-c^2)  ]_-cosδ^cosδ

 
We have 
    e^ i w_1c/ i w_1ℱ(arccos c)  1/ 2 π√(1-c^2)  ]_-cosδ^cosδ≤2/ 2 πw_1 sinδℱ_∞
 since we may bound the value at -cosδ and cosδ separately.

Let  
    𝒢(θ)  =  dlogℱ(θ)/dθ =  ∑_m=1^∞(mθ + π/2) · v_m + i∑_m=2^∞(m θ + π/2)· w_m.

 
 We have 
    d/dc    (   ℱ( arccos c)   1/ 2 π√(1-c^2)) = ℱ(arccos c) 1/ 2 π√(1-c^2)(  c/1-c^2 -𝒢(arccos c) /√(1-c^2))
 
Respectively applying (<ref>) and reversing the change of variables c= cosθ, then applying Cauchy-Schwarz, and finally using the integral ∫_δ^π -δ1/sin^2 θ dθ = 2 cosδ/sinδ gives

    ∫_- cosδ^cosδ e^ i  w_1c/ i w_1d/dc ( ℱ( arccos c)   1/ 2 π√(1-c^2)) dc
    
    =    ∫_δ^π-δ e^ i  w_1cosθ/ i w_1ℱ (θ)  (  cosθ/sin^2 θ + 𝒢(θ) /sinθ)    dθ/2π
    ≤   ℱ_∞/w_1∫_δ^π - δcosθ/sin^2 θdθ/2π + ℱ_∞/w_1√(∫_δ^π -δ1/sin^2 θdθ/2π·∫_δ^π-δ𝒢(θ)^2 dθ/2π)
    ≤   cosδℱ_∞/w_1πsinδ + ℱ_∞/w_1√(cosδ/πsinδ∫_δ^π-δ|𝒢(θ) |^2 dθ/2π).

 Plugging (<ref>) and (<ref>) into (<ref>), and then applying (<ref>), we obtain
 
    | ∫_0^π    e^ i w_1cosθℱ(θ)  dθ/2π| ≤δℱ_∞/π + ℱ_∞/πw_1sinδ + cosδ/w_1πsinδ + ℱ_∞/w_1√(cosδ/πsinδ∫_δ^π-δ𝒢(θ)^2 dθ/2π) .

 
 A symmetrical argument gives

    | ∫_π^2π  e^ i w_1cosθℱ(θ)  dθ/2π| ≤δℱ_∞/π + ℱ_∞/πw_1sinδ + cosδℱ_∞/w_1πsinδ + 1/w_1√(cosδ/πsinδ∫_π + δ^2π-δ𝒢(θ)^2 dθ/2π) .

 
We have 
    ∫_δ^π-δ𝒢(θ)^2dθ/2π  + ∫_π+ δ^2π-δ𝒢(θ)^2dθ/2π≤∫_0^2π𝒢(θ)^2 dθ/2π  =1/2∑_m=1^∞v_m^2 +  1/2∑_m=2^∞w_m^2
 with the last equality using the definition (<ref>) of 𝒢. Combining (<ref>), (<ref>), and (<ref>) gives

    | ∫_0^2π  e^ i w_1cosθℱ(θ)   dθ/2π| /ℱ_∞


    ≤2 δ/π + 2/πw_1sinδ + 2 cosδ/w_1πsinδ + 1/w_1√(cosδ/πsinδ ( ∑_m=1^∞v_m^2+ ∑_m=2^∞w_m^2))


    ≤2 δ/π + 1/w_1δ + 2 /w_1πδ + 1/w_1√( 1/πδ( ∑_m=1^∞v_m^2+ ∑_m=2^∞w_m^2)).

Taking δ= 1/√(w_1) we obtain (<ref>).
 
 
 | We have
  
    ∑_m=1^∞v_m/m≤ζ(2)/4 +  ∑_m=1^∞min ( v_m^2, v_m/m)  .
 
   For each m we have  v_m/m ≤v_m^2 + 1/4m^2 by completing the square and thus  v_m/m ≤min ( v_m^2, v_m/m)   + 1/4m^2. Summing over m gives the statement.  

 
 
  There exists δ_3>0, constants C_1, C_2>0, and a function 𝒮 [0,∞] → [0,1] such that the following hold:

 Let (v_m)_m=1^∞ and (w_m)_m=1^∞ be sequences of vectors in ℝ^2. If ∑_m=1^∞ m v_m^2+ ∑_m=1^∞w_m^2 < ∞ then

    ∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m + i∑_m=1^∞(m θ) · w_m/m dθ/2π
    ≤    e^v_1^2/4 - δ_3 min ( v_1^4, v_1^2) + ∑_m=2^∞min (C_1  v_m^2, v_m/m )  (  1 + C_2 ∑_m=2^∞w_m^2 + C_2 ∑_m=2^∞v_m^2 ) 𝒮( w_1 ) .


Furthermore, we have 
    𝒮(y) = O ( 1/√(y)),
 we have 
    𝒮(y) ≤  e^  - y^2/4 + O( y^3),
 and 𝒮(y) is bounded away from 1 for y in each fixed closed interval not containing 0.



Here the function 𝒮 describes how much savings is obtained in our estimate from cancellation induced by w_1. The advantage of writing the bound in this way is we can treat 𝒮(w_1) as a single quantity for calculations that are uniform in w_1 but also easily break up into different ranges.

From this point on, we always take 𝒮 to be a function as in <ref>.

  Take C_1 as in <ref>. Fix δ_3, C_2 to be chosen later. We choose δ_3 sufficiently small and C_2 sufficiently large. Neither depends on the other. We will always write a fixed closed interval not containing zero as [C_3,C_4]. (There is also no relation between C_3,C_4 and the other variables.) We let
 
    S(v_1,…, w_1,…) = ∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m + i∑_m=1^∞(m θ) · w_m/m dθ/2π/  e^v_1^2/4 - δ_3 min ( v_1^4, v_1^2) + ∑_m=2^∞min (C_1  v_m^2, v_m/m )  (  1 + C_2 ∑_m=2^∞w_m^2 + C_2 ∑_m=2^∞v_m^2 ) .

and define

    𝒮(y) = inf_ (v_m)_m=1^∞, (w_m)_m=1^∞∈ (ℝ^2)^ℕ
    w_1 =y   S(v_1,…, w_1,…)

so that (<ref>) holds by definition and the upper bounds on 𝒮(y) can be checked by checking corresponding upper bounds on S(v_1,…, w_1,…). That is, for (<ref>) it suffices to have 
    S(v_1,…, w_1,…) = O ( 1/√(w_1)),
 for (<ref>) it suffices to have 
    S(v_1,…, w_1,…)  ≤ e^ - w_1^2/4 + O(w_1^3)
 and for 𝒮(y) to be bounded away from 1 for y in an interval [C_3,C_4] not containing 0 it suffices to have S(v_1,…, w_1,…) bounded away from 1 for w_1∈ [C_3,C_4]

First note that we can always bound the integral  ∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m + i∑_m=1^∞(m θ) · w_m/m dθ/2π by its untwisted form ∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m  dθ/2π which is bounded by Proposition <ref>
as 
    e^v_1^2/4 - δ_2 min ( v_1^4, v_1^2) + ∑_m=2^∞min (C_1  v_m^2, v_m/m ) ) ,
 which can be bounded by

    e^v_1^2/4 - δ_3 min ( v_1^4, v_1^2) + ∑_m=2^∞min (C_1  v_m^2, v_m/m ) ) 1/ 1+ δ_3 v_1^4
 since  log (1+ δ_3 v_1)^4 ≤ (δ_2 -δ_3) min ( v_1^4, v_1^2) for δ_3 sufficiently small with respect to δ_2.

It follows that 
    S(v_1,…, w_1,…)  ≤1/   (1 + δ_3 v_1^4) ( 1 + C_2 ∑_m=2^∞w_m^2 + C_2 ∑_m=2^∞v_m^2 )  .
 This in particular implies that S(v_1,…, w_1,…) ≤ 1 and thus 𝒮(y) ≤ 0. Since 𝒮(y) is clearly nonnegative we see that 𝒮 is indeed a function from [0,∞] to [0,1].


Furthermore, as long as  δ_3 v_1^4 + C_2 ∑_m=2^∞ w_m^2  + C_2 ∑_m=2^∞v_m^2 ≥w_1^2/4 we have  S(v_1,…, w_1,…) ≤1/ 1+ w_1^2/4.  Since 1/ 1+ w_1^2/4 is O (1/√(w_1)), is equal to e^ - w_1^2/4+ O(w_1^3), and is bounded away from 1 for C_3 ≤w_1≤ C_4, for the remainder of the argument we may assume that

    δ_3 v_1^4 + C_2 ∑_m=2^∞ w_m^2  + C_2 ∑_m=2^∞v_m^2 < w_1^2/4

which notably implies

    ∑_m=1^∞ w_m^2  + ∑_m=1^∞v_m^2 < O ( w_1^2) + O(1)

since δ_3 v_1^4  ≥ C_2 v_1^2 + O(1)  and w_1^2= O(w_1^2).


First we check (<ref>). Since S(v_1,…,w_1,…) ≤ 1, it suffices to check (<ref>) for y sufficiently small.  Note that (<ref>) implies that, as long as w_1 is sufficiently small, v_1 is as small as desired, and the same holds for ∑_m=2^∞v_m^2 + ∑_m=2^∞w_m^2. By Cauchy-Schwarz

    v_1+ w_1+ ∑_m=2^∞√(v_m^2 + w_m^2)/m  ≤v_1+w_1 + √( (ζ(2)-1)  ∑_m=2^∞( v_m^2 + w_m^2))

which we can take to be as small as desired, in particular ensuring the assumption of <ref> is satisfied, and we have

    log∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m + i∑_m=1^∞(m θ) · w_m/mdθ/2π
 

    ≤v_1^2/4 - w_1^2/4 - δ_3 v_1^4 + O_δ_3  ( v_1^6 + w_1^3 + ∑_m=2^∞ (v_m^2+ w_m^2 ))

and by (<ref>) we have v_1^6 = O ( w_1^3) so that term can be ignored. Thus (<ref>) is less than or equal to 

    e^v_1^2/4- δ_3 v_1^4/ e^v_1^2/4 - δ_3 min ( v_1^4, v_1^2)e^ O(  ∑_m=2^∞ (v_m^2+ w_m^2 ))/1 + C_2 ∑_m=2^∞w_m^2 + C_2 ∑_m=2^∞v_m^2   e^ - w_1^2/4 + O ( w_1^3 ) .
 We have dropped the term ∑_m=2^∞min (C_1  v_m^2, v_m/m ) from the denominator as it is always ≥ 1 but is unneeded.

We clearly have e^v_1^2/4- δ_3 v_1^4/ e^v_1^2/4 - δ_3 min ( v_1^4, v_1^2)≤ 1  and we have e^ O(  ∑_m=2^∞ (v_m^2+ w_m^2 ))/1 + C_2 ∑_m=2^∞w_m^2 + C_2 ∑_m=2^∞v_m^2≤ 1 as long as w_1 is sufficiently small and C_2 is sufficiently large since we can bound e^x by 1 + cx for any c>1 as long as x is sufficiently small.

Thus (<ref>) is less then or equal to e^ - w_1^2/4 + O ( w_1^3 ) for w_1 sufficiently small which gives 𝒮(y) ≤  e^ - y^2/4 + O ( y^3) for y sufficiently small, and thus for all y, verifying (<ref>).


  
   Next we check (<ref>). Before applying <ref>, we observe that 
   
    sup_θ∈ [0,2π]   e^∑_m=1^∞(m θ) · v_m/m ≤  e^∑_m=1^∞v_m/m≤ e^ζ(2)/4 + ∑_m=1^∞min( v_m^2, v_m/m)
 
    ≪   e^v_1^2/4 - δ_3 min ( v_1^4, v_1^2) + ∑_m=2^∞min (C_1  v_m^2, v_m/m )
 since δ_3 ≤1/4 and C_1≥ 1.
   Next observe that (using (<ref>) to handle the case w_1 small in the first inequality)
   
    4/π + 1 +   √(∑_m=1^∞v_m^2 +   ∑_m=2^∞w_m^2)/√(π)w_1^1/4≪  1  +  √(∑_m=1^∞v_m^2 +   ∑_m=2^∞w_m^2)
 
    ≤5/4 + ∑_m=1^∞v_m^2 +  ∑_m=2^∞w_m^2  ≪  1 + δ_3 v_1^4 + C_2  ∑_m=2^∞w_m^2 + C_2 ∑_m=2^∞v_m^2 .
 Putting these bounds together with <ref>, we obtain

    ∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m + i∑_m=1^∞(m θ) · w_m/m dθ/2π
 
    ≤sup_θ∈ [0,2π]   e^∑_m=1^∞(m θ) · v_m/m /√(w_1)( 4/π + 1 +   √(∑_m=1^∞v_m^2 +  ∑_m=2^∞w_m^2)/√(π)w_1^1/4)


    ≪  e^v_1^2/4 - δ_3 min ( v_1^4, v_1^2) + ∑_m=2^∞min (C_1  v_m^2, v_m/m )1/√(w_1) ( 1 + δ_3 v_1^4 + C_2  ∑_m=2^∞w_m^2 + C_2 ∑_m=2^∞v_m^2 )


    ≤  e^v_1^2/4 - δ_3 min ( v_1^4, v_1^2) + ∑_m=2^∞min (C_1  v_m^2, v_m/m )1/√(w_1)  (1 + δ_3 v_1^4)   ( 1 + C_2  ∑_m=2^∞w_m^2 + C_2 ∑_m=2^∞v_m^2 )

which verifies (<ref>).
   
   
  
  Next we consider w_1 in an interval I= [C_3,C_4] not containing 0. Applying <ref> and then <ref> we have

    S(v_1,…, w_1,…) ≤∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m + i∑_m=1^∞(m θ) · w_m/m dθ/2π/∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m dθ/2π
 
    ≤ 1- e^ -2 ∑_m=1^∞v_m/mmin(  ∑_m=1^∞w_m^2/π^2 m^2, 1/ 2∑_m=1^∞w_m^2).

Since 
    ∑_m=1^∞v_m/m ≤√(ζ(2) ∑_m=1^∞v_m^2)≪w_1≤ C_4=O(1)
 and 
    ∑_m=1^∞w_m^2  ≪w_1^2 ≤ C_4^2 =O(1)

and 
    ∑_m=1^∞w_m^2/π^2 m^2≥w_1^2/π^2m^2≥ C_3^2/π^2 m^2,
 we see that (<ref>) is at most 1-ϵ for some ϵ>0. Hence S(v_1,…, w_1,…) ≤ 1-ϵ for w_1∈ [C_3, C_4], as desired. 


 Let (v_m)_m=1^∞ and (w_m)_m=1^∞ be sequences of vectors in ℝ^2. If ∑_m=1^∞ m v_m<∞ and ∑_m=1^∞ w_m^2 < ∞ then

    ∫_0^2π   e^∑_m=1^∞(m θ) · v_m/m + i∑_m=1^∞(m θ) · w_m/m dθ/2π
 
    ≤ e^v_1^2/4 - δ_3 min ( v_1^4, v_1^2)  𝒮( w_1)   ∏_m=2^∞( e^min ( C_1 v_m^2, v_m/m )  (1 + C_2 w_m^2 )( 1+ C_2 v_m^2) ).
 

 This follows from taking <ref> and separating terms, using the trivial bound

    1 + C_2 ∑_m=2^∞w_m^2 + C_2 ∑_m=q^∞v_m^2 ≤∏_m=2^∞ (1 + C_2 w_m^2 ) (1+ C_2 v_m^2) .





Write A_n for ∑_d| n, d<n E_d and B_n for ∑_d| n, d<n E_d d/n.
 Let v_1,…, v_k and w_1,…, w_k be vectors in ℝ^2. Then
    𝔼 [ e^∑_n=1^k X_n,ξ· v_n  +  i ∑_n=1^k X_n,ξ· w_n  ] 
    ≤   ∏_n=1^k  (  e ^ E_n  v_n^2/4  +  min (C_1 A_n v_n^2, B_n  v_n  )  - δ_3 E_n min ( v_n^4, v_n^2)  (1 + C_2 v_n^2)^ A_n   (1+ C_2 w_n^2)^ A_n𝒮(w_n ) ^E_n).


 Taking <ref>, using <ref> to bound each factor, and then rearranging terms, we obtain

    𝔼 [ e^∑_n=1^k X_n,ξ· v_n  +  i ∑_n=1^k X_n,ξ· w_n  ]
 
    ≤∏_n=1^k ( (   e^v_n^2/4 - δ_3 min ( v_n^4, v_n^2)  𝒮( w_n ) )^ E_n∏_ d| n, d<n( e^min ( C_1 v_n^2, v_nd/n )  (1 + C_2 w_n^2 )(1+ C_2 v_n^2) )^ E_d) .
 
Using 
    ∑_ d| n, d<n E_d min ( C_1 v_n^2, v_nd/n ) ≤min (C_1 ∑_ d| n, d<n E_d v_n^2, ∑_d| n E_d v_n d/n )
 we obtain (<ref>). 

The following facts about A_n, B_n, and E_n will be useful in the remainder of the argument.

 We have the following identities and inequalities for A_n, B_n, E_n:
 
    B_n + E_n = q^n/n.

 
    A_n = O ( q^n/2/n) .

  
    B_n = O ( q^n/2/n).

  
    E_n = q^n/n + O( q^n/2/n) .


    E_n> 2A_n+4  as long as  q>5 .

There exists a positive constant c such that 

    E_n/2- 2 A_n - q/2-1> c(A_n + E_n)   as long as  q>11 and  n>1.



 For (<ref>) by definition we have B_n+ E_n = ∑_d| n E_d d/n and ∑_d| n  E_d  d= q^n by either counting elements of the finite field 𝔽_q^n or a zeta function argument. This in particular implies E_n ≤q^n/n, which we will use repeatedly in the remaining proofs. 

For (<ref>), we observe that the largest possible d satisfying d| n and d<n is n/2 and every other solution is at most n/3, so  

    A_n=∑_d| n, d<n E_d ≤  E_n/2 + ∑_d ≤ n/3 E_n/3≤ q^n/2/ (n/2) + ∑_d≤ n/3 q^d = O(q^n/2/n) + O(q^n/3)= O(q^n/2/n).


(<ref>) follows from (<ref>) upon observing B_n≤ A_n. 

(<ref>) follows from (<ref>) and (<ref>). 

To obtain  (<ref>) and (<ref>) we redo the above proofs with explicit constants to prove the inequalities for all q^n sufficiently large and then use exact formulas to handle the finitely many remaining possible values of (q,n). 

 The proof of (<ref>) gives
 
    A_n=∑_d| n, d<n E_d ≤  E_n/2 + ∑_d ≤ n/3 E_n/3≤ q^n/2/ (n/2) + ∑_d≤ n/3 q^d≤ 2 q^n/2/n + 1/1-q^-1 q^n/3 .


For X> 0 we have 2X^1/6> 1/1-7^-1log_7 X so we have 
    2q^n/6 > 1/1-7^-1log_7 q^n ≥1/1-q^-1log_q q^n =  1/1-q^-1 n
  so

    A_n ≤ 4 q^n/2/n.
 
Then we have

    B_n = A_n ≤ 4  q^n/2/n

and

    E_n= q^n/n-B_n ≥q^n/n - 4 q^n/2/n .


Thus (<ref>) is satisfied as long as  
    q^n/n - 4q^n/2/n  >  4 q^n/2/n +4
 i.e. as long as  
    1 >  8 q^-n/2  +4 n q^-n
 which by (<ref>) follows from 
    1> 8 q^-n/2 + 8 q^-5n/6
 which holds for q^n> 95.2. Because q>7 this holds for all n>2. But for n=1 we have E_n=q and A_n=0 so (<ref>) becomes q>4 which is satisfied for all q>5 and for n=2 we have E_n = q^2-q/2 and A_n=q so (<ref>) becomes q^2-5q/2 >4 which is satisfied for all q>5.
 
 
For (<ref>) it follows from (<ref>), (<ref>), and (<ref>) that E_n/2- 2 A_n - q/2-1 = q^n/(2n) + O(q^n/2/n) +O(q) and A_n+ E_n =q^n/n + O(q^n/2/2) so (<ref>) is satisfied for q^n sufficiently large. Thus it suffices to prove 
    E_n/2- 2 A_n - q/2-1> 0
  as then we can choose c small enough to ensure (<ref>) is satisfied for the finitely many remaining values of q,n.  Then by (<ref>), (<ref>), and (<ref>) it suffices to prove

    q^n/2n - 2 q^n/2/n - 8 q^n/2/n - q^1+ n/6/n-  2 q^n/6/n>0

or equivalently

    1/2 > 10 q^-n/2  + q^ 1 - 5n/6  + 2 q^- 5n/6.

which since n ≥ 2 follows from

    1/2 > 10 q^-n/2  + q^ -n/3  + 2 q^- 5n/6

which holds for q^n>697.4. Because q>11 this holds for all n>2. For n=2 we have E_n = q^2-q/2 and A_n=q so (<ref>) becomes q^2-11q/4-1 >0 which is satisfied for all q>11.  

 For any n>0 and v∈ℝ^2, we have 
    e ^ - B_n  v^2/4 +  min (C_1 A_n v^2, B_n  v  )  - δ_3 E_n min ( v^4, v^2)  (1 + C_2 v^2)^ A_n  ≤ e^O ( min ( q^n/2v^2 /n, 1/n)) .


  We note first that  min (C_1 A_n v^2, B_n  v  ) = O ( A_n v^2) and also log (1 + C_2 v^2)^ A_n  = O ( A_n v^2) so the left-hand side of (<ref>) is always e^ O ( A_n v^2) = e^ O ( q^n/2v^2/n) by (<ref>).

We next check that the left-hand side of (<ref>) is ≤ e^ O(1/n). First in the range v≤ 1, it suffices to check that   e^ O ( A_n v^2) - δ_3 E_n  v^4 ≤ e^ O(1/n) but the exponent is ≤ O ( A_n^2/δ_3 E_n) and therefore is ≤ O(1/n) by  (<ref>) and (<ref>). For v≥ 1, it suffices to check that  e^ O ( A_n v^2) - δ_3 E_n v^4≤  e^ O(1/n) which is automatic as long as  O(A_n) - δ_3 E_n ≤ 0 which happens for all but finitely many n, again by  (<ref>) and (<ref>). For these finitely many n, it suffices to check that (<ref>) goes to 0 as n goes to ∞, which is clear as e^min ( C_1 A_n v^2, v) is merely exponential in a  linear function of v while (1 + C_2 v^2)^ A_n is polynomial and these are both dominated by e^ -δ_3 E_n min ( v^4, v^2) which is exponential in a quadratic function.


 Let v_1,…, v_k and w_1,…, w_k be vectors in ℝ^2. Then
    𝔼 [ e^∑_n=1^k X_n,ξ· v_n  +  i ∑_n=1^k X_n,ξ· w_n  ] ≤∏_n=1^k  (  e ^q^n/nv_n^2/4   + O ( min ( q^n/2v_n^2 /n, 1/n))(1+ C_2 w_n^2)^ A_n𝒮(w_n ) ^E_n).

 
 This follows from plugging E_n = q^n/n-B_n from (<ref>) into <ref> and then plugging in <ref>. 




 §.§ Pointwise bounds


Recall that F(x_1,…, x_k) is the probability density function of X_1,ξ,…, X_n,ξ.

 Assume q>5. Let x_1,…, x_k be vectors in ℝ^2.   Then 

    F(x_1,…, x_k)/∏_n=1^k ( e^ -  nx_n^2/q^nn/q^n π)≤  O(  e^ O ( ∑_n=1^k min (  n q^-3n/2x_n^2, n^-1 ) ) ).


 Let  
    v_n = 2 n x_n  /q^n
 for all n from 1 to k.  By the Fourier inversion formula we have 

    (2π)^2k e^∑_n=1^k x_n · v_n  F( x_1,…, x_n) = ∫_w_1,…, w_k ∈ℝ^2  e^ -i ∑_n=1^k x_n · w_n 𝔼 [ e^∑_n=1^k X_n,ξ· v_n  +  i ∑_n=1^k X_n,ξ· w_n  ] dw_1 … dw_k .

(<ref>) and <ref>, give

    (2π)^2k e^∑_n=1^k x_n · v_n  F( x_1,…, x_n) ≤∫_w_1,…, w_k ∈ℝ^2𝔼 [ e^∑_n=1^k X_n,ξ· v_n  +  i ∑_n=1^k X_n,ξ· w_n  ] dw_1 … dw_k


    ≤∏_n=1^k e^q^n/nv_n^2/4+ O ( min ( q^n/2v_n^2 /n, 1/n))∫_w_1,…, w_k ∈ℝ^2∏_n=1^k  (1+ C_2 w_n^2)^ A_n𝒮( w_n )^E_n   dw_1 … dw_k


    = ∏_n=1^k e^q^n/nv_n^2/4+  O ( min ( q^n/2v_n^2 /n, 1/n))∏_n=1^k ∫_w∈ℝ^2 (1+ C_2 w^2)^ A_n𝒮( w )^E_n dw.


We first tackle the inner integral 
    ∫_w∈ℝ^2  (1+ C_2 w^2)^ A_n𝒮( w )^E_n   dw.
 First note that for w large, we have (1+ C_2 w^2)^ A_n = O(w^2A_n) and S( w )^E_n = O ( w^- E_n/2), so for the integral to converge, it is necessary and sufficient to have E_n/2 > A_n+2, which follows from (<ref>).

Since we can absorb the integrals (<ref>) for small n into the implicit constant, we will be focused on the asymptotic evaluation of (<ref>) for large n.

For w bounded we have 
    log (1+ C_2 w^2) = 1+ C_2 w^2 + O (w^4)
 and 
    log𝒮(w)  ≤w^2/4 + O( w)^3
 so

    (1+ C_2 w^2)^ A_n𝒮(w)^E_n≤  e^(C_2 A_n - E_n/4) w^2 +O ( q^n w^3/n)
  (using (<ref>) and (<ref>))  which for w≤ (q^n/n)^-2/5 is ≤ e^(C_2 A_n - E_n/4) w^2   e^ ( q^n/n)^-1/5 so the integral over w≤ (q^n/n)^-2/5 is bounded by e^ ( q^n/n)^-1/5 times

    ∫_w∈ℝ^2 e^ (C_2 A_n - E_n/4) w^2 dw =  π/E_n/4 - C_2 A_n  = π/q^n/4n - O ( q^n/2 /n)  =4π n /q^n + O (  n/q^3n/2)

(for n large), with the e^ ( q^n/n)^-1/5 factor itself contributing an error term of size O (  (q^n/n)^-6/5).

The integral (<ref>) over w> (q^n/n)^-2/5 will give additional error terms.

First in the range where w> (q^n/n)^-2/5 but w is bounded by a fixed small constant, we have  (C_2 A_n - E_n/4) w^2= - ( q^n/4n+ O (q^n/2/n)) w^2 which is larger by a constant factor than O ( q^n w^3/n), so the integrand of (<ref>) in this range is at most e^ - c q^n w^2/n≤ e^ - c (q^n/n)^1/5 for a small constant c. Since the area of this range is O(1), this range gives an error term of size decreasing superexponentially in q^n/n.

For w greater than a large fixed constant R, we have 1 + C_2 w^2 = O( w)^2and 𝒮(w) = O ( w^-1/2). This gives a bound of O(1)^A_n + E_nw^ 2A_n - E_n/2 for the integrand or O(1)^ A_n + E_n   R^ 2+ 2A_n - E_n/2 for the integral (<ref>), and since both A_n + E_n and E_n -4 A_n -4 are asymptotic to q^n/n by (<ref>) and (<ref>), we can choose R large enough that the second term dominates and the error term decays superexponentially in q^n/n.

For the intermediate range of w between two fixed constants, we also get superexponential decay simply by observing that (1 + C_2 w^2)^2= O(1)^A_n and 𝒮(w)^E_n≤ (1-ϵ)^E_n for some ϵ>1, while A_n = o(E_n) and E_n increases exponentially by (<ref>) and (<ref>), so the integrand has superexponential decay and the length of the integral on this range is O(1).

So all these error terms are dominated by the  O (  (q^n/n)^-6/5), giving 

    ∫_w∈ℝ^2  (1+ C_2 w^2)^ A_n𝒮(w)^E_n dw = 4π n /q^n +O (  (q^n/n)^-6/5)

which implies 

    ∏_n=1^k ∫_w∈ℝ^2 (1+ C_2 w^2)^ A_n𝒮(w)^E_n   dw = O ( ∏_n=1^k 4 π n/q^n) .


Since v_n = 2 n x_n / q^n we have 
    v_n · x_n =   q^n/ 4nv_n^2 + n/q^nx_n^2
 and plugging (<ref>) and (<ref>) into (<ref>) we obtain

    (2π)^2k e^∑_n=1^k (q^n/ 4nv_n^2 + n/q^nx_n^2 )  F( x_1,…, x_n) ≤∏_n=1^k  (  e ^q^n/nv_n^2/4  + O( min ( q^n/2v_n^2 /n, 1/n)) ) O ( ∏_n=1^k 4 π n/q^n )

and solving for F(x_1,…, x_n)  gives

    F( x_1,…, x_n) ≤ O(1)   ∏_n=1^k  (  e ^O( min ( q^n/2v_n^2 /n, 1/n))) ∏_n=1^k  ( (2π)^-2 e^ - n/q^nx_n^24 π n/q^n)


    ≪ e^ O ( ∑_n=1^k min ( q^n/2v_n^2/n, 1/n))∑_n=1^k  ∏_n=1^k  ( e^ - n/q^nx_n^2n /π q^n).

Plugging in the definition (<ref>) of v_n and dividing by ∏_n=1^k  ( e^ - n/q^nx_n^2n /π q^n) gives (<ref>). 





 Assume q>5. Let x_1,…, x_k be vectors in ℝ^2.   Then 

    F(x_1,…, x_k)/∏_n=1^k ( e^ -  nx_n^2/q^nn/q^n π)≤  O( k^O(1) ) = O(N^O(1)).


 We have  
    ∑_n=1^k min (  n q^-3n/2x_n^2, n^-1 ) ≤∑_n=1^k n^-1 = O(log k)
 and plugging (<ref>) into (<ref>), together with the trivial bound k≤ N, gives (<ref>). 



 §.§ Hermite polynomial expansion bounds


The goal of this subsection is to prove a formula, Corollary <ref>, for F(x_1,…, x_k) as the product of a Gaussian probability density function times a sum of Hermite polynomials weighted by certain coefficients h_a_1,1,…, a_k,2, together with bounds for the coefficients h_a_1,1,…, a_k,2. The bounds for the coefficients will start in <ref> with a complicated bound expressed in terms of an integral and conclude in <ref> which bounds a sum of squares of the  h_a_1,1,…, a_k,2 which exactly equals the error, in L^2 norm integrated against the Gaussian measure, of a low-degree polynomial approximation for F(x_1,…,x_k)/∏_n=1^k ( e^ -  nx_n^2/q^nn/q^n π) obtained using these Hermite polynomials. We begin with a brief review of Hermite polynomials.

The (probabilist's) Hermite polynomials are defined as:

    𝐻𝑒_n ( x) = (-1)^n e^x^2/2d^n/ dx^n e^ - x^2/2

and their key property is the orthogonality when integrated against the Gaussian measure

    ∫_-∞^∞𝐻𝑒_n ( x)  𝐻𝑒_m ( x)   e^ - x^2/2/√(2π) dx =  n!    if  n=m 
     0    if  n ≠ m .


After a change of variables, this implies for any σ>0

    ∫_-∞^∞𝐻𝑒_n ( x/σ) 𝐻𝑒_m ( x/σ)    e^ - x^2/2σ^2/√(2π)σ dx =  n!    if  n=m 
     0    if  n ≠ m .






We have

    ∫_-∞^∞  e^ i xy𝐻𝑒_n ( x)    e^ - x^2/2/√(2π) dx= ∫_-∞^∞  e^ i xy (-1)^n ( d^n/ dx^ne^ - x^2/2/√(2π))  dx =  ∫_-∞^∞(  d^n/ dx^ne^ i xy)  e^ - x^2/2/√(2π)  dx = y^n e^y^2/2

and a change of variables gives

    ∫_-∞^∞ e^ i xy𝐻𝑒_n ( x/σ)    e^ - x^2/2σ^2 /√(2π)σ dx=  σ^n y^n e^σ^2 y^2/2.


Now we introduce the notation that will be needed for our first bound on the coefficients.

Let C_5 be the implicit constant in the big O in <ref>.   Let n be a positive integer. 

For a a positive integer and r a positive real number, write 
    ℐ_n(a,r)=  1/π∫_-∞^∞ e^   C_5 min ( q^n/2v ^2 /n, 1/n)/ v+ i  r^a+1 dv .
 For a=0, write 
    ℐ_n(a,r)=1.


For a_n,1,a_n,2 two nonnegative integers, let

    ℒ_n (a_n,1,a_n,2)= inf_r_n,1, r_n,2≥ 0 
     r_n,j=0  if and only if  a_n,j=0  (1+ C_2 (r_n,1^2+r_n,2^2))^A_n𝒮(√( r_n,1^2+r_n,2^2 ))^E_n/ e^ - q^n/nr_n,1^2 + r_n,2^2 /4ℐ_n(a_n,1,r_n,1) ℐ_n(a_n,2,r_n,2).





For w_n∈ℝ^2, write w_n,1 for its first coordinate and w_n,2 for its second coordinate.



  There exists a tuple (h_a_1,1,…, a_k,2)_ a_1,1,…, a_k,2∈ℤ^≥ 0 of complex numbers indexed by 2k-tuples of nonnegative integers such that 
    𝔼 [ e^  i ∑_n=1^k X_n,ξ· w_n  ]  = e^ - ∑_n=1^k q^n/nw_n^2/4∑_ a_1,1, …, a_k,2∈ℤ^≥ 0 h_a_1,1,…, a_k,2∏_n=1^k (w_n,1^a_n,1 w_n,2^a_n,2)

and for each tuple a_1,1,…, a_k,2 of nonnegative integers we have

    h_a_1,1,…, a_k,2≤∏_n=1^k  ℒ_n ( a_n,1, a_n,2 ).
 

We will shortly see in <ref> that the h_a_1,1,…, a_k,2 are the coefficients of an expansion of F by Hermite polynomials. The remaining results in this subsection will be devoted to proving more straightforward upper bounds on the h_a_1,1,…, a_k,2, culminating in <ref> which bounds a certain sum of h_a_1,1,…, a_k,2 which will appear in the proof of <ref>.

 The fact that h_a_1,1,…, a_k,2 exist and the sum is absolutely convergent follows from the fact that 
    𝔼 [ e^  i ∑_n=1^k X_n,ξ· w_n  ] /  e^ - ∑_n=1^k q^n/nw_n^2/4
 is an entire function which is clear as the random variables X_n,ξ are bounded so the numerator is entire while the denominator is entire and nowhere vanishing.

To estimate the coefficients, we use the Cauchy integral formula. We explain the argument in detail only in the case that a_1,1,…,a_k,2 are all nonzero. The general case follows the same ideas, but is notationally more complicated.

 For f a function of a complex variable z, which is bounded on loci in the complex plane where the imaginary part of z is bounded, the coefficient of z^a in the Taylor expansion at 0 of f is given for any r>0 by 

    1/2π( ∫_ r i + ∞^r i - ∞ f(z) dz/z^a+1 + ∫_ -ri- ∞^-ri +∞ f(z) dz/z^a+1)

or writing z= x+iy, by

    1/2π( ∫_ -∞^∞ f(x+ ir ) dx/ (x+ir)^a+1 + ∫_ -∞^∞ f(x-ir ) dx/(x-ir)^a+1) .
 On the other hand, for a=0, the value is simply f(0).

Thus for f a function of complex variables z_1,1, …, z_k,2 which is bounded on loci in ℂ^2k where the imaginary parts of all coordinates are bounded, the coefficient of ∏_n=1^k ( z_n,1^a_n,1 z_n,2^a_n,2) in f is given for any r_1,1,…, r_k,2>0 by

    1/(2π)^2k∑_ϵ_1,1,…, ϵ_n,k∈± 1∫_ℝ^2k f( x_1,1+  i ϵ_1,1 r_1,1,…, x_k,2 + i ϵ_k,2 )   dx_1,1… d x_k,2/∏_n=1^k ( (x_n,1+  i ϵ_n,1 r_n,1)^a_n,1+1 (x_n,2+  i ϵ_n,2 r_n,2)^a_n,2+1 ) .

If some of the a_n,j are 0, we can drop the corresponding variables x_n,j from the integral as well as drop the sums over ϵ_n,j and a corresponding number of factors of (2π).  We apply this to the function of  u_1,…, u_k∈ℂ^2, with u_n = v_n + i w_n, given by

    𝔼 [ e^∑_n=1^k X_n,ξ· u_n  ]  / e^ - ∑_n=1^k q^n/n u_n · u_n/4

to obtain

    i^ -∑_n=1^k (a_n,1 + a_n,2) h_a_1,1,…, a_k,2
 
    = 1/(2π)^2k∑_ϵ_1,1,…, ϵ_k,2∈± 1∫_v_1,…, v_n∈ℝ^2𝔼 [ e^∑_n=1^k X_n,ξ· v_n + i ∑_n=1^k X_n,ξ·w̃_n  ]  / e^∑_n=1^k q^n/nv_n^2 + 2 i v_n·w̃_n - w̃_n^2/4 dv_1… dv_k/∏_n=1^k (( v_n,1 + i ϵ_n,1 r_n,1)^a_n,1+1 ( v_n,2 + i ϵ_n,2 r_n,2)^a_n,2+1 )

where w̃_n = ( ϵ_n,1 r_n,1, ϵ_n,2 r_n,2). Taking absolute values and applying <ref> gives

    h_a_1,1,…, a_k,2
 
    ≤1/(2π)^2k∑_ϵ_1,1,…, ϵ_k,2∈± 1∫_v_1,…, v_n∈ℝ^2𝔼 [ e^∑_n=1^k X_n,ξ· v_n + i ∑_n=1^k X_n,ξ·w̃_n  ]/ e^∑_n=1^k q^n/nv_n^2 - w̃_n^2/4 dv_1… dv_k/∏_n=1^k (  v_n,1 + i  r_n,1^a_n,1+1 v_n,2 + i  r_n,2^a_n,2+1)


    ≤1/(2π)^2k∑_ϵ_1,1,…, ϵ_n,k∈± 1∫_v_1,…, v_n∈ℝ^2∏_n=1^k  (  e ^q^n/nv_n^2/4  + C_5 min ( q^n/2v_n^2 /n, 1/n)    (1+ C_2 w̃_n^2)^ A_n𝒮(w̃_n)^E_n) dv_1… dv_k / e^∑_n=1^k q^n/nv_n^2 - w̃_n^2/4∏_n=1^k (  v_n,1 + i  r_n,1^a_n,1+1 v_n,2 + i  r_n,2^a_n,2+1)


    =  1/(2π)^2k∑_ϵ_1,1,…, ϵ_k,2∈± 1∏_n=1^k   (1+ C_2 w̃_n^2)^ A_n𝒮(w̃_n)^E_n/ e^ - q^n/nw̃_n^2/4∫_v_1,…, v_n∈ℝ^2∏_n=1^k e ^  C_5  min ( q^n/2v_n^2 /n, 1/n)   dv_1… dv_k /∏_n=1^k (  v_n,1 + i  r_n,1^a_n,1+1 v_n,2 + i  r_n,2^a_n,2+1) .

For w_n=  (r_n,1, r_n,2), we have w̃_n=w_n and since w̃_n only appears in (<ref>) via its absolute value, we may simplify by replacing w̃_n by w_n and then removing the sum over ϵ_n,j, obtaining

    h_a_1,1,…, a_k,2≤1/π^2k∏_n=1^k   (1+ C_2 w_n^2)^ A_n𝒮(w_n)^E_n/ e^ - q^n/nw_n^2/4∏_n=1^k∫_v_n∈ℝ^2 e^  C_5 min ( q^n/2v_n^2 /n, 1/n) / v_n,1 + i  r_n,1^a_n,1+1 v_n,2 + i  r_n,2^a_n,2+1 dv_n.

If some of the a_n,j are 0, we drop the corresponding variables v_n,j from the integral in (<ref>) as well as a corresponding number of factors of π.

We can bound e^   C_5  min ( q^n/2v_n^2 /n, 1/n) by e^   C_5  min ( q^n/2v_n,1^2 /n, 1/n)  e^ C_5 min ( q^n/2v_n,2^2 /n, 1/n) so the inner integral splits as a product

    ∫_v_n∈ℝ^2 e^   C_5 min ( q^n/2v_n^2 /n, 1/n) / v_n,1 + i  r_n,1^a_n,1+1 v_n,2 + i  r_n,2^a_n,2+1 dv_n ≤∏_j=1^2∫_-∞^∞ e^  C_5  min ( q^n/2v_n,j^2 /n, 1/n) / v_n,j + i  r_n,j^a_n,j+1 dv_n,j
 which matches π^2 ℐ_n ( a_n,1 , r_n,1) ℐ_n (a_n,2,r_n,2), giving a bound of ∏_n=1^k  ℒ_n ( a_n,1, a_n,2 ) once we choose r_n,1,r_n,2 to be values where the minimum in the definition of ℒ_n is attained (or comes arbitrarily close to being attained). If some of the variables are 0, since we drop the integral and the factor of π, we obtain 1, again maching the definition of ℐ_n ( a_n,j , r_n,j ). Here we use the fact that e^   C_5  min ( q^n/2v_n,1^2 /n, 1/n)  =1 if v_n,j=0 . 




We are now ready to state our Hermite expansion. The proof relies on <ref> below, but there is no circularity as Corollary <ref> is not used until the next section – we state it here for motivation.
  For (h_a_1,1,…, a_k,2)_ a_1,1,…, a_k,2∈ℤ^≥ 0  as in <ref> we have
 
    F(x_1,…, x_k) = ∏_n=1^k ( e^ -  nx_n^2/q^nn/q^n π)  ∑_ a_1,1, …, a_k,2∈ℤ^≥ 0 h_a_1,1,…, a_k,2∏_n=1^k  He_a_n,1( x_n,1√(2n/q^n)) He_a_n,2(x_n,2√(2n/q^n) )  √(2n/q^n)^ a_n,1+ a_n,2.
 
 
  We
 take Fourier transforms of both sides. Using <ref> to compute the Fourier transform of the left-hand side and (<ref>) to compute the Fourier transform of the right-hand side, we see the Fourier transforms are equal. the The left-hand side is an L^2 function by <ref> and the right-hand side is L^2 by <ref> below and <ref>). By invertibility of the Fourier transform for L^2 functions, both sides are equal. 

We are now ready to estimate the inegral ℐ_n and local terms ℒ_n. The easiest, but most important, estimate is the following:

  We have 
    ℒ_n(0,0)≤ 1
 for all n. 

 We set r_n,1=r_n,2=0 and all the terms are manifstly equal to 1 in this case, except for 𝒮(0), which is ≤ 1 by <ref>. (In fact one can also check 𝒮(0)=1 using <ref> but we don't need this.) 

For (a_n,1,a_n,2 ) ≠ (0,0) it will suffice to bound ℒ_n(a_n,1,a_n,2) to within a constant factor. To that end, we have the following bound for ℐ_n:

  We have 
    ℐ_n(r,a) ≪1/ r^a √(a+1)
 where we adopt the convention 0^0=1. 

 The case a=0 is 1 ≪ 1 which is clear. For a>0 convexity of the logarithm gives the lower bound

    v+ i  r^a+1 = ( v^2 + r^2 )^a+1/2  = r^a+1(v^2/r^2 + 1)^a+1/2≥ r^ a+1( 1 + a+1/2v^2/r^2).

We also have 
    e^   C_5 min ( q^n/2v ^2 /n, 1/n)≤ e^C_5/n≪ 1.
 Thus

    ℐ_n( a,r) ≪∫_-∞^∞1 / v+ i  r^a+1 dv  ≤∫_-∞^∞1/ r^a+1( 1 + a+1/2v^2/r^2) dv.
 The change of variables x= √(a+1/2)v/r gives

    ∫_-∞^∞1/ r^a+1( 1 + a+1/2v^2/r^2) dv= 1/r^a√(2/a+1)∫_∞^∞1/1+x^2≪1/ r^a √(a+1).



This allows us to prove the following general bound, where we have reintroduced w_n for compactness of notation.

 Fix integers n>0 and a_n,0, a_n,1≥ 0. Let r_n,1,r_n,2 be nonnegative real numbers such that r_n,j=0 if and only if a_n,j=0. Let w_n=(r_n,1,r_n,2). Then

    ℒ_n (a_n,1, a_n,2) ≪ (1+ C_2 w_n^2)^A_n𝒮( w_n )^E_n/ e^ - q^n/nw_n^2 /4  a_n,1^r_n,1 a_n,2^r_n,2√(a_n,1+1 )√(a_n,2+1).


 This follows from the definition of ℒ_n, after observing that a minimum is bounded by its value at any point, applying <ref> to bound ℐ_n(a_n,j, r_n,j), and substituting w_n for √(r_n,1^2+r_n,2^2). 


We will specialize Lemma <ref> at different values of a_n,1,a_n,2 in different ranges. First we give a lemma helpful for a_n,1+a_n,2 small:



 For integers n>0 and a_n,1,a_n,2≥ 0 we have 

    ℒ_n(a_n,1, a_n,2 )  ≪ e^ O ( a_n,1 + a_n,2)( q^n/n)^a_n,1+ a_n,2/3  a_n,1^ - a_n,1/3 a_n,2^- a_n,2/31/√(a_n,1+1)√(a_n,2 +1)   .


  We apply Lemma <ref> and set r_n,j=  √(na_n,j/q^n) . We  have 
    𝒮( w_n ) ≤ e^ - w_n^2/4 + O (w_n^3)
 and 
    1 + C_2 w_n^2 ≤ e^ C_2 w_n^2
 so

    (1+ C_2 w_n^2)^ A_n𝒮(w_n)^E_n/ e^ - q^n/nw_n^2/4≤ e^ A_n C_2 w_n^2 + q^n/nw_n^2/4 - E_n w_n^2/4 + O ( E_n w_n^3)
        =  e^ A_n C_2 w_n^2 + B_n  w_n^2/4 + O ( E_n w_n^3) = e^ O ( q^n/2/nw_n^2 + q^n/nw_n^3 )  = e^ O (  q^n/nw_n^3 )
  since if a_n,1=a_n,2=0 the exponent is 0 and otherwise w_n≥√(n/q^n)≥ q^-n/2.
We have 
    w_n^3 = (r_n,1^2 + r_n,2^2)^3/2 = n/q^n  (  a_n,1^2/3 + a_n,2^2/3)^3/2  = O ( n/q^n (a_n,1 + a_n,2 ) ).
 
Combining (<ref>) and (<ref>), we have

    (1+ C_2 w_n^2)^ A_n𝒮(w_n)^E_n/ e^ - q^n/nw_n^2/4 r_n,1^a_n,1 r_n,2^a_n,2 =  e^ O ( a_n,1 + a_n,2)/ r_n,1^a_n,1 r_n,2^a_n,2 =e^ O ( a_n,1 + a_n,2)( q^n/n)^a_n,1+ a_n,2/3  a_n,1^ - a_n,1/3 a_n,2^- a_n,2/3 .
 Adding the √(a_n,1+1)√(a_n,2+1) term, we obtain the statement.
 

Next we give a lemma for  a_n,1+a_n,2 large.


 There exists an absolute constant C_6 such that for integers n>0 and a_n,1,a_n,2≥ 0 with  a_n,1 + a_n,2 larger than C_6 q^n/n, we have

    ℒ_n(a_n,1, a_n,2 )  ≪ O(1)^ A_n+E_n( q^n/2n)^E_n/4- A_n+ a_n,1+ a_n,2/2 (a_n,1+a_n,2)^ A_n - E_n/4  e^ a_n,1 + a_n,2/2a_n,1^ - a_n,1/2 a_n,2^ - a_n,2/21/√(a_n,1+1)√(a_n,2 +1)   .
 

  We apply Lemma <ref> and set r_n,j= = √(2n/q^n a_n,j)  We have 
    w_n = √( r_n,1^2 + r_n,2^2) = √(2n/q^n  (a_n,1 + a_n,2))
 which is larger than an absolute constant, so 
    1+ C_2 w_n^2= O ( w_n^2) = O (  2n/q^n  (a_n,1 + a_n,1) )
 and 
    𝒮( w_n) = O ( w_n^-1/2 ) = O ( ( q^n/2n)^1/4 (a_n,1+a_n,2)^-1/4)
 while 
    e^ - q^n/nw_n^2/4 = e^ -   a_n,1 + a_n,2/2
 and 
    r_n,j^a_n,j = ( q^n/2n)^ - a_n,j/2  a_n,j^a_n,j/2 .
 Putting (<ref>), (<ref>), (<ref>), and (<ref>) all together gives

    (1+ C_2 w_n^2)^ A_n𝒮(w_n)^E_n/ e^ - q^n/nw_n^2/4 r_n,1^a_n,1 r_n,2^a_n,2 = O(1)^ A_n+E_n( q^n/2n)^E_n/4- A_n + a_n,1+ a_n,2/2 (a_n,1+a_n,2)^ A_n - E_n/4  e^ a_n,1 + a_n,2/2a_n,1^ - a_n,1/2 a_n,2^ - a_n,2/2 .
 Adding the √(a_n,1+1)√(a_n,2+1) term, we obtain the statement.


Our final lemma will be used for a_n,1+a_n,2 in an intermediate range.

 For each C_7, there exists ϵ>0 such that for integers n>0 and a_n,1,a_n,2≥ 0 with a_n,1 + a_n,2 less than C_7 q^n/n, we have

    ℒ_n(a_n,1, a_n,2 )  ≪  e^( 1/2 - ϵ) (a_n,1+ a_n,2)( q^n/2n)^a_n,1 + a_n,2/2 a_n,1^ - a_n,1/2 a_n,2^ - a_n,2/21/√(a_n,1+1)√(a_n,2 +1)   .
  

 For small values of n, there are finitely many possibilities and it suffices to apply <ref>, absorbing any discrepancies into the implicit constant, so we may assume n is large. We apply Lemma <ref> and set r_n,j= = √(2n/q^n a_n,j).  We have w_n <C_7.  There exists ϵ>0 such that 𝒮( x) < e^ -   ϵ x^2 for all x ≤ C_7 (since any ϵ< 1/4 works for x sufficiently small and some ϵ works on every bounded interval away from 0). We furthermore have 

    (1+ C_2 w_n^2)^ A_n𝒮(w_n)^E_n≤ e^ C_2 A_n w_n^2 - ϵ E_n w_n^2  = e^ C_2 A_n w_n^2 + ϵ B_n w_n^2 -  ϵq^n/nw_n^2 ≤ e^ -  ϵ/2q^n/nw_n^2
 for n is sufficiently large. We also have

    r_n,j^a_n,j = ( q^n/2n)^ - a_n,j/2  a_n,j^a_n,j/2 .
  Using (<ref>) and (<ref>), we have

    (1+ C_2 w_n^2)^ A_n𝒮(w_n)^E_n/ e^ - q^n/nw_n^2/4 r_n,1^a_n,1 r_n,2^a_n,2≤  e^( 1/4 - ϵ/2) q^n/nw_n^2 ( q^n/2n)^a_n,1 + a_n,2/2  a_n,1^ - a_n,1/2 a_n,2^ - a_n,2/2
 

    =  e^( 1/2 - ϵ) (a_n,1+ a_n,2)( q^n/2n)^a_n,1 + a_n,2/2  a_n,1^ - a_n,1/2 a_n,2^ - a_n,2/2 .
 Adding the √(a_n,1+1)√(a_n,2+1) term, we obtain the statement.

We are now ready to give our final bound we need on the h_a_1,1,…, a_k,2s.

 If q>11 then for (h_a_1,1,…, a_k,2)_ a_1,1,…, a_k,2∈ℤ^≥ 0  as in <ref> we have

    ∑_ a_1,1, …, a_k,2∈ℤ^≥ 0
    ∑_n=1^k n (a_n,1+ a_n,2) > kh_a_1,1,…, a_k,2^2 ∏_n=1^k a_n,1! a_n,2! ( 2n/q^n)^ a_n,1+ a_n,2≪_q  k ^ - q-2/2 .
  

 It suffices to prove


    ∑_ a_1,1, …, a_k,2∈ℤ^≥ 0( min(k, ∑_n=1^k n (a_n,1+ a_n,2) )^q/2h_a_1,1,…, a_k,2^2 ∏_n=1^k a_n,1! a_n,2! ( 2n/q^n)^ a_n,1+ a_n,2  = O_q(k)  .


The inequality ∑_n=1^k  n (a_n,1+ a_n,2) ≤∏_n=1^k (1 +  n (a_n,1+ a_n,2) ) and (<ref>) gives

    ∑_ a_1,1, …, a_k,2∈ℤ^≥ 0( min (k, ∑_n=1^k n (a_n,1+ a_n,2) ))^q/2h_a_1,1,…, a_k,2^2 ∏_n=1^k a_n,1! a_n,2! ( 2n/q^n)^ a_n,1+ a_n,2


    ≤∑_ a_1,1, …, a_k,2∈ℤ^≥ 0∏_n=1^k ( min(k,  1+  n (a_n,1+ a_n,2) ))^q/2h_a_1,1,…, a_k,2^2 ∏_n=1^k a_n,1! a_n,2! ( 2n/q^n)^ a_n,1+ a_n,2


    ≪∑_ a_1,1, …, a_k,2∈ℤ^≥ 0∏_n=1^k  ( ( min(k,  1+  n (a_n,1+ a_n,2)) )^q/2ℒ_n ( a_n,1, a_n,2)    a_n,1! a_n,2! ( 2n/q^n)^ a_n,1+ a_n,2)


    =   ∏_n=1^k  ( ∑_a_n,1, a_n,2 =0^∞( min(k,  1+  n (a_n,1+ a_n,2) ))^q/2ℒ_n ( a_n,1, a_n,2)    a_n,1! a_n,2! ( 2n/q^n)^ a_n,1+ a_n,2)
 so it suffices to show 

    ∑_a_n,1, a_n,2 =0^∞( min(k,  1+  n (a_n,1+ a_n,2)) )^q/2ℒ_n ( a_n,1, a_n,2)    a_n,1! a_n,2! ( 2n/q^n)^ a_n,1+ a_n,2 =  O(k)     if  n=1 
     1 + O_q(1/n^2)     if  n>1 .

Removing the a_n,1,a_n,2=0 term handled by <ref>, this is equivalent to

    ∑_a_n,1, a_n,2∈ℤ^≥ 0
     (a_n,1,a_n,2)≠ (0,0) ( min (k,  n (a_n,1+ a_n,2)) )^q-2/2ℒ_n ( a_n,1, a_n,2)    a_n,1! a_n,2! ( 2n/q^n)^ a_n,1+ a_n,2 =   O(k)     if  n=1 
      O_q(1/n^2)     if  n>1 .

Stirling's formula gives 

    a_n,j! ≪ a_n,j^a_n,j e^ - a_n,j√(a_n,j+1).


We split a_n,1 + a_n,2 into three ranges.  We fix constants C_8 sufficiently small and C_9 sufficiently large. For a_n,1+ a_n,2≤ C_8 q^n/n^7 we apply <ref> to bound ℒ_n ( a_n,1, a_n,2). For a_n,1+ a_n,2≥ C_9 q^n/n we apply <ref>. For a_n,1,a_n,2∈ (C_8 q^n/n^7, C_9 q^n/n) we apply <ref>. 

Applying <ref> and (<ref>) to the terms in (<ref>) with a_n,1+ a_n,2≥ C_9 q^n/n, we obtain


    ∑_a_n,1, a_n,2∈ℤ^≥ 0
     a_n,1+ a_n,2≥ C_9 q^n/n( min(k,   n (a_n,1+ a_n,2)) )^q/2ℒ_n ( a_n,1, a_n,2)    a_n,1! a_n,2! ( 2n/q^n)^ a_n,1+ a_n,2


    ≪∑_a_n,1, a_n,2∈ℤ^≥ 0
     a_n,1+ a_n,2≥ C_9 q^n/n( min(k,   n (a_n,1+ a_n,2)) )^q/2 O(1)^ A_n+E_n( q^n/2n)^E_n/2- 2A_n (a_n,1+a_n,2)^ 2A_n - E_n/21/√(a_n,1+1)√(a_n,2 +1)


    ≪∑_a ∈ℤ^≥ 0
     a ≥ C_9 q^n/n( min(k,   n a) )^q/2 O(1)^ A_n+E_n( q^n/2n)^E_n/2- 2A_n  a^ 2A_n - E_n/2.


We now handle the cases n=1 and n>1 separately. For n=1, we have E_n=q and A_n=0 and the terms depending on q and n may be absorbed into the implicit constant. This gives

    ∑_a ∈ℤ^≥ 0
     a ≥ C q ( min (k,    a) )^q/2  a^ - q/2 .


The terms where a ≤ k contribute at most ∑_ a=1^k 1 =k and the remaining terms contribute k^q/2∑_a=k+1^∞  a^- q/2 =O(k ), so this indeed gives O(k).

For n>1 and at all subsequent points in the argument, we will ignore the min(k , · ). This gives 

    ∑_a ∈ℤ^≥ 0
     a ≥ C_9 q^n/n  n^q/2  O(1)^ A_n+E_n( q^n/2n)^E_n/2- 2A_n  a^ 2A_n - E_n/2 + q/2


    ≪ n^q/2  O(1)^ A_n+E_n( q^n/2n)^E_n/2- 2A_n( C_9 q^n/n)^ 2 A_n - E_n/2+ q/2 +1


    = n^q/2  O(1)^ A_n+E_n( C_9 )^ 2 A_n - E_n/2+ q/2+1 ( q^n/n)^q/2+1.

Since q> 11,  by (<ref>), the exponent E_n/2- 2 A_n - q/2-1 is greater than a constant multiple of A_n + E_n so choosing C_9 sufficiently large the ( C_9 )^ 2 A_n - E_n/2+ q/2 term dominates O(1)^A_n+E_n by a factor that is doubly exponential in n. Since ( q^n/n)^q/2 and n^q-2/2 are at most singly exponential in n, they are easily dominated and the product is O(1/n^2). So indeed (<ref>) is O(1/n^2) for n>1 and O( k) for n=1.

Applying <ref> and (<ref>) to the terms in (<ref>) with a_n,1+ a_n,2∈ (C_8 q^n/n^7, C_9 q^n/n), we obtain 

    ∑_a_n,1, a_n,2∈ℤ^≥ 0
     a_n,1+ a_n,2∈ ( C_8 q^n/n^7,  C_9 q^n/n ) ( n (a_n,1+ a_n,2)  )^q/2ℒ_n ( a_n,1, a_n,2)    a_n,1! a_n,2! ( 2n/q^n)^ a_n,1+ a_n,2


    ≪∑_a_n,1, a_n,2∈ℤ^≥ 0
     a_n,1+ a_n,2∈ ( C_8 q^n/n^7,  C_9 q^n/n ) ( n (a_n,1+ a_n,2)  )^q/2 e^ - 2ϵ  (a_n,1+ a_n,2)1/√(a_n,1+1)√(a_n,2 +1)


    ≤∑_a_n,1, a_n,2∈ℤ^≥ 0
     a_n,1+ a_n,2∈ ( C_8 q^n/n^7,  C_9 q^n/n ) ( n C_9 q^n/n)^q/2 e^-2 ϵ C_8 q^n/n^7


    ≪(q^n/2)^2 ( n C_9 q^n/n)^q/2 e^-2 ϵ C_8 q^n/n^7

and the term e^-2 ϵ C_8 q^n/n^7 decreases doubly exponential in n while the remaining terms increase singly-exponentially so the product decreases doubly-exponentially and in particular is O(1/n^2). 

Applying <ref> and (<ref>) to the terms in (<ref>) with a_n,1+ a_n,2≤ C_8 q^n/n^7, we obtain

    ∑_a_n,1, a_n,2∈ℤ^≥ 0
     a_n,1+ a_n,2∈ (0,  C_8 q^n/n^7]  ( n (a_n,1+ a_n,2)  )^q/2ℒ_n ( a_n,1, a_n,2)    a_n,1! a_n,2! ( 2n/q^n)^ a_n,1+ a_n,2


    ≪∑_a_n,1, a_n,2∈ℤ^≥ 0
     a_n,1+ a_n,2∈ (0,  C_8 q^n/n^7]   e^ O ( a_n,1 + a_n,2)( q^n/n)^-  a_n,1+ a_n,2/3  a_n,1^a_n,1/3 a_n,2^a_n,2/31/√(a_n,1+1)√(a_n,2 +1)


    ≤∑_a_n,1, a_n,2∈ℤ^≥ 0
     a_n,1+ a_n,2∈ (0,  C_8 q^n/n^7]   e^ O ( a_n,1 + a_n,2)( q^n/n)^-  a_n,1+ a_n,2/3  (C_8 q^n/n^7) ^a_n,1/3 (C_8 q^n/n^7) ^a_n,2/3


    = ∑_a_n,1, a_n,2∈ℤ^≥ 0
     a_n,1+ a_n,2∈ (0,  C_8 q^n/n]    ( C_8^ 1/3 e^O(1) n^-2 )^ a_n,1 + a_n,2≤∑_a_n,1, a_n,2∈ℤ^≥ 0
     a_n,1+ a_n,2  >0    ( C_8^ 1/3 e^O(1) n^-2 )^ a_n,1 + a_n,2 .

We have 
    ∑_a_n,1, a_n,2∈ℤ^≥ 0
     a_n,1+ a_n,2  >0   x^ a_n,1 + a_n,2 =  2x-x^2/ (1-x)^2  = O(x)
 for x sufficiently small so, taking C_8 sufficiently small, this is O ( C_8^1/3 e^O(1) n^-2 ) = O(n^-2), as desired. 




 



§ THE CHIMERA


In this section, we prove <ref>, <ref>, and <ref>. The proof of <ref> is direct and independent of the prior results. To prove <ref> and <ref>, we combine estimates from the previous section on the function F with estimates from the literature on the measure μ_rm.

  It suffices to check first that the support of μ_ch is contained in the support of μ_rm and second that the support of μ_ch is contained in the support of μ_ep.

Since μ_rm is the pushforward of the Haar measure on U(N), and the support of Haar measure is all of U(N), the support of μ_rm is (the closure of) the image of U(N). Since μ_ch is also the pushforward of a measure on U(N), its support is also contained in (the closure of) the image of U(N) and hence in the support of  μ_rm.

For a point L^* ∈ to be contained in the support of μ_ep, each neighborhood of L^* must contain a random Euler product L_ξ with positive probability. Since the topology is the product topology, a basis for the neighborhood consists of the sets of power series in q^-s whose first n coefficients are all within ϵ of the first n coefficients of L^*. Whether L_ξ lies in this neighborhood only depends on ξ(𝔭) for 𝔭 of degree ≤ n. The set of functions from 𝔭 of degree ≤ n to the circle a finite-dimensional manifold, the uniform measure on this manifold is supported everywhere, and the map from this to the first n coefficients of L_ξ, so it suffices for L^* to be in the image of this manifold. In other words, it suffices to check there is a single function ξ from primes to the unit circle such that first n coefficients of L_ξ agree with the first n coefficients of L^*.

If L^* lies in the support of μ_ch then there is certainly a function ξ where the first k coefficients of L_ξ agree with L^* since  L^*=  ( I - q^1/2 - s  M) for some M with 
    F( - q^1/2(M),…, -q^k/2 (M^k)/k)>0
 where F is the probability density function of the first k coefficients of the logarithm of a random L_ξ. Since F( - q^1/2(M),…, -q^k/2 (M^k)/k)>0, the density is nonzero, so there exists ξ such that the first k coefficients of log L_ξ  match - q^1/2(M),…, -q^k/2 (M^k)/k.

We now check by induction on n that for each n ≥ k there exists a function ξ_n such that the first n coefficients of  log L_ξ_n match  - q^1/2(M),…, -q^n/2 (M^n)/n. For n=k this is what we just checked. For n=k, we choose ξ_n so that ξ_n(f) =ξ_n-1(f) for all f of degree <n, so that the first n-1 coefficients of L_ξ_n match the first n-1 coefficients of L_ξ_n-1. The nth coefficient of log L_ξ is then 

    ∑_𝔭∈𝔽_q[u]^+ 
    irreducible
    𝔭=nξ_n(𝔭) + ∑_𝔭∈𝔽_q[u]^+ 
    irreducible
    𝔭| n 
    𝔭≠ n ξ_n(𝔭)/ (n/𝔭)
 so it suffices to choose ξ_n so that

    ∑_𝔭∈𝔽_q[u]^+ 
    irreducible
    𝔭=nξ_n(𝔭) = -q^n/2 (M^n)/n -  ∑_𝔭∈𝔽_q[u]^+ 
    irreducible
    𝔭| n 
    𝔭≠ n ξ_n(𝔭)/ (n/𝔭)

We can choose ∑_𝔭∈𝔽_q[u]^+ 
irreducible
𝔭=nξ_n(𝔭) to be any complex number of absolute value ≤  E_n so it suffices to check the right hand side has absolute value ≤ E_n. We have 
    ∑_𝔭∈𝔽_q[u]^+ 
    irreducible
    𝔭| n 
    𝔭≠ n ξ_n(𝔭)/ (n/𝔭) = O( q^n/2)
 and 
    q^n/2 (M^n)/n≤ q^n/2 N/n
  

Since E_n is greater than a constant times q^n/n, the right hand side is is ≤ E_n as long as q^n/2 > O(N + n) which happens as long as  n/ log N is sufficiently large. Since n>k, this happens if k/log N is sufficiently large, and since k = ⌊ N^β⌋≥⌊ N^1/4⌋, this happens for N sufficiently large.


For this section, a convenient coordinate system for  consists of the variables b_n defined so that b_n(L) is √(n/q^n) times the coefficient of q^-ns in log L, so that 
    L = e^∑_n=1^∞√(q^n/n) b_n(L) q^-s.
 Thus 
    b_n(L_ξ ) = √(n/q^n)X_n,ξ
 and because the definition (<ref>) of L_M gives 
    L_M (s)=  ( I - q^1/2 - s  M) = e^∑_n=1^∞ q^n/2 q^-ns(M^n)/n  ,
 we have 
    b_n(L_M) = - ( M^n)/√(n) .


Recall the measures μ_ep and μ_rm on .  We define another measure μ_g on  as the unique measure where the b_n are independent complex standard normal random variables and the constant coefficient is 1. The utility of μ_g for our purposes is that it serves as an approximation for both μ_ep and μ_rm.

In particular, define a projection map ηℂ[[q^-s ]] →ℂ^k that sends L to b_1(L),…, b_k(L).

The next two results give strong estimates, in different forms, comparing μ_rm and μ_g. 

<cit.> For any β<1/2, setting k = ⌊ N^β⌋, as long as N is sufficiently large in terms of β, the total variation distance between the pushforward measures η_* μ_rm^N and η_* μ_g is 
    ≤ e^- (1- o_N(1)) N^1-βlog (N^1-β)
 where o_N(1) goes to 0 as N goes to ∞ for any fixed β. 

 This is a restatement of <cit.>, with a simplified but weaker bound. We explain how our notation compares. 

It is immediate from the definitions that η_* μ_g is a product of k independent standard complex Gaussian distributions. Taking the real and imaginary parts, and multiplying by -√(2), we obtain a product of 2k independent standard real Gaussians, exactly what is called 𝐆 in <cit.>. (Their m is our k).

Similarly, η_* μ_rm is the distribution of b_1(M),…, b_k(M) =  -(M^1)/√(1),…, -  (M^k)/√(k). Taking real and imaginary parts and multiplying by - √(2), this is exactly the distribution called 𝐗 in <cit.>.

The total variation bound follows from <cit.>, noting that the factor 1.4 · 10^-13 n^3β-3/2 is ≤ 1 and can be ignored. 


<cit.> Let ϕ∈ℂ[ c_0, c_1,…, c_0,c_1,…] have degree ≤ N. Then 
    ∫_ϕμ_g = ∫_ϕμ_rm^N.
 

We use the total variation distance between measures to control differences between integrals against the measures in a couple different ways:

 For probability measures μ_1,μ_2 with total variation distance δ and a function f, we have

    ∫ f μ_1 - ∫ f μ_2≤  2 δsupf


    ∫ f μ_1 - ∫ f μ_2≤√(δ)( √(∫f^2 μ_1) + √(∫f^2 μ_2))
 

By the definition of total variation distance, we can write μ_1 - μ_1'  =μ_2 -μ_2' where μ_1' ≤μ_1 and μ_2'≤μ_2 are measures with total mass δ. We obtain (<ref>) by noting the integral of f against any measure is bounded by the sup-norm of f times the total mass of that measure.  We obtain (<ref>) by applying Cauchy-Schwarz to f and the density functions μ_1'/μ_1 and μ_2'/μ_2, which are 1-bounded and integrate to δ and thus have L^2 norm at most √(δ). 

We repeatedly use the following lemma to compare integrals against different measures:

 Let G be a measurable function of complex variables b_1,…, b_k. We have

    ∫_U(N)   G( - (M),…, - (M^k)/√(k) ) μ_Haar =  ∫_  G( b_1(L),…, b_k(L))  μ_rm = ∫_ℂ^k  G(b_1,…,b_k)  η_* μ_rm.
 
and

    ∫_  G(b_1(L),…, b_k(L)) μ_ep = ∫_ℂ^k  G(b_1,…,b_k) η_* μ_ep = ∫_ℂ^k  G(b_1,…,b_k)  F( q^1/2 b_1 ,…,  q^k/2  b_k/ √(k) )/∏_j=1^k  ( e^  -  |b_j |^2  j / q^j π ) η_* μ_g .




 We prove (<ref>) first. Both equalities follow from the fact that the integral of a function against the pushforward of a measure is the integral of the pullback against the measure. For the first equality, we also use that μ_rm is the pushforward of μ_Haar along M ↦ L_M by definition, and use (<ref>) to compute the pullback of G along M ↦ L_M. 

We now prove (<ref>). The first equality is similar to (<ref>). For the second equality, we use  that the probability density function of η_* μ_g, in the variables b_1,…, b_k, is  ∏_j=1^k  ( e^  -  b_j ^2 1 /π), so we have

    ∫_ℂ^k   G(b_1,…, b_k)  F( q^1/2 b_1 ,…,  q^k/2  b_k/ √(k) )/∏_j=1^k  ( e^  -  |b_j |^2  j / q^j π ) η_* μ_g = ∫_ℂ^k   G(b_1,…, b_k)  F( q^1/2 b_1 ,…, -q^k/2  b_k/ √(k) )/∏_j=1^k  (   j / q^j )  db_1 … db_k.


Because F is the probability density function of the tuple of random variables X_1,ξ,…, X_k,ξ, and by (<ref>) we have b_n(L_ξ) = √(n/q^n)X_n,ξ, it follows that F( q^1/2 b_1 ,…, -q^k/2  b_k/ √(k) )/∏_j=1^k  (   j / q^j ) is the probability density function of the tuple of random variables b_1(L_ξ),…, b_k(L_ξ), i.e. of the measure η_* μ_ep, giving

    ∫_ℂ^k   G(b_1,…, b_k)  F( q^1/2 b_1 ,…, -q^k/2  b_k/ √(k) )/∏_j=1^k  (   j / q^j )  db_1 … db_k=  ∫_ℂ^k  G(b_1,…,b_k) η_* μ_ep.
  






The first step to proving the main theorems is to evaluate γ:

  If q>5 then for N sufficiently large in terms of β, 
    ∫_U(N) F( -q^1/2(M),…, -q^k/2 (M^k)/k)/∏_j=1^k  ( e^  -   (M^j)^2 /  j   j q^j / π) μ_Haar = 1 + O ( e^ -(1- o_N(1)) N^1-βlog (N^1-β)  ) .
 
In other words, the γ in the definition (<ref>) of μ_weighted is 1 + O ( e^ -(1- o_N(1)) N^1-βlog (N^1-β)  ) for N sufficiently large. 

  (<ref>) gives

    ∫_U(N) F( - q^1/2(M),…, -q^k/2 (M^k)/k)/∏_j=1^k  ( e^  -   (M^j)^2 /  j  j / q^j π) μ_Haar= ∫_ℂ^k F(  q^1/2 b_1 ,…, q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  |b_j| ^2   j / q^j π) η _* μ_rm.


Similarly, (<ref>) gives

    ∫_ℂ^k F(  q^1/2 b_1 ,…, q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  |b_j| ^2   j / q^j π) η _* μ_g = ∫_ℂ^k 1 μ_ep = 1 .


Next we will prove 
    ∫_ℂ^k F(  q^1/2 b_1 ,…, q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  |b_j| ^2   j / q^j π) η _*μ_rm  - F(  q^1/2 b_1 ,…, q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  |b_j| ^2   j / q^j π) η _* μ_g =  O ( e^ -(1- o_N(1)) N^1-βlog (N^1-β)) .
 To check (<ref>), we apply (<ref>). We use <ref> to bound the total variation distance by e^- (1- o_N(1)) N^1-βlog (N^1-β). We apply <ref> to bound the sup-norm by O(N^O(1)), and note that the O(N^O(1)) can be absorbed into the o_N(1) in the exponent.

Combining (<ref>), (<ref>), and (<ref>), we obtain (<ref>).


The main part of proving <ref> is the following:


 Assume that q>5. Let ϕ∈ℂ[ c_1,c_2,…, c_1, c_2,…] have degree ≤ k. For N sufficiently large in terms of β, we have

    ∫_ F(  q^1/2 b_1 ,…, q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  |b_j| ^2   j / q^j π) ϕμ_g = ∫_ϕμ_ep
 and

    ∫_ϕμ_ep  =  ∫_ F(  q^1/2 b_1 ,…, q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  |b_j| ^2   j / q^j π) ϕμ_rm  + O (e^ -(1/2 - o_N(1)) N^1-βlog (N^1-β)ϕ_2 ) .



 First note that, since ϕ has degree ≤ k, we can express ϕ as a polynomial only in terms of c_1,c_2,…, c_k, c_1 , c_2,…, c_k. Since c_1,…, c_k can be expressed as polynomials in b_1,…,b_k, it follows that ϕ can be expressed as a polynomial in b_1,…,b_k, b_1,…, b_k, which we will refer to as ϕ̃. Then we have

    ∫_ϕμ_ep = ∫_ℂ^k F(  q^1/2 b_1 ,…, q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  |b_j| ^2   j / q^j π) ϕ̃η _* μ_g
 by (<ref>),

    ∫_ F(  q^1/2 b_1 ,…, q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  |b_j| ^2   j / q^j π) ϕμ_g = ∫_ℂ^k F(  q^1/2 b_1 ,…, q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  |b_j| ^2   j / q^j π) ϕ̃η _* μ_g

by compatibility of integration with pushforward of measures, and

    ∫_ F(  q^1/2 b_1 ,…, q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  |b_j| ^2   j / q^j π) ϕμ_rm =∫_ F(  q^1/2 b_1 ,…, q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  |b_j| ^2   j / q^j π) ϕ̃η _* μ_rm

by (<ref>).

Combining (<ref>) and (<ref>), we obtain (<ref>). From (<ref>) and (<ref>), we see that to prove (<ref>), it suffices to prove that

    ∫_ℂ^k F(  q^1/2 b_1 ,…, q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  |b_j| ^2   j / q^j π) ϕ̃η _* μ_g = ∫_ F(  q^1/2 b_1 ,…, q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  |b_j| ^2   j / q^j π) ϕ̃η _* μ_rm + O (e^- (1/2 - o_N(1)) N^1-βlog (N^1-β)ϕ_2 ) .

We will do this by applying (<ref>) to η _* μ_rm and η _* μ_g. We have by <ref> and <ref>

    ∫_ℂ^k F(  q^1/2 b_1 ,…, q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  |b_j| ^2   j / q^j π) ^2 |ϕ̃|^2 η _* μ_g
    ≪ N^O(1)   ∫_ℂ^k  |ϕ̃|^2 η _* μ_g = N^O(1)∫_ϕ^2 μ_g= N^O(1)∫_ϕ^2 μ_rm = N^O(1)ϕ

and an identical argument, except skipping the <ref> step, gives

    ∫_ℂ^k F(  q^1/2 b_1 ,…, q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  |b_j| ^2   j / q^j π) ^2 |ϕ̃|^2 η _* μ_rm≪ N^O(1)ϕ .

Plugging (<ref>), (<ref>), and <ref> into <ref>, we obtain 

    ∫_ℂ^k F(  q^1/2 b_1 ,…, q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  |b_j| ^2   j / q^j π) ϕ̃η _* μ_g
 
    = ∫_ F(  q^1/2 b_1 ,…, q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  |b_j| ^2   j / q^j π) ϕ̃η _* μ_rm + O (N^O(1) e^ -(1/2 - o_N(1)) N^1-βlog (N^1-β)ϕ_2 )
 which, absorbing the N^O(1) into the o_N(1), gives (<ref>). 

For the next two proofs,  we observe that for ϕ∈ℂ[c_0,c_1,…, c_0,c_1,…] we have

    ∫_ϕμ_ch  =  ∫_U(N)ϕ(L_M)    μ_weighted  = γ∫_U(N) ϕ(L_M)     F( -q^1/2(M),…, -q^k/2 (M^k)/k)/∏_j=1^k  ( e^  -   (M^j)^2 /  j   j q^j / π) μ_Haar
 by the definition (<ref>) of μ_weighted.
  



We have 
    ∫_U(N) ϕ(L_M)     F( -q^1/2(M),…, -q^k/2 (M^k)/k)/∏_j=1^k  ( e^  -   (M^j)^2 /  j   j q^j / π) μ_Haar  = ∫_ϕ F(  -q^1/2 b_1 ,…, -q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  b_j^2   j / q^j π) μ_rm
     =      ∫_ϕμ_ep  +  O (e^ -(1/2 - o_N(1)) N^1-βlog (N^1-β)ϕ_2 )
 by (<ref>) and <ref>.  Combining (<ref>) and (<ref>) gives exactly the main term and error term of (<ref>) except with an extra factor of γ. Using <ref> to estimate γ, 
we see that multiplying by γ introduces an additional error term of size  ∫_ϕμ_ep O( e^- (1 - o_N(1) )N^1-βlog (N^1-β) ). But 

    ∫_ϕμ_ep≤∫_ϕμ_ep = ∫_ϕ F(  -q^1/2 b_1 ,…, -q^k/2  b_k / √(k) )/∏_j=1^k  ( e^  -  b_j^2   j / q^j π) μ_g≤ O ( k^O(1)  ) ∫_ϕμ_g≤
 
    O ( k^O(1)  ) √(∫_ϕ^2 μ_g)= O ( k^O(1)  ) √(∫_ϕ^2 μ_rm)= O( k^O(1)) ϕ_2
 by the trivial bound, (<ref>), <ref>, Cauchy-Schwarz, <ref>, and definition, so this error term can be absorbed into the O (e^ -(1/2 - o_N(1)) N^1-βlog (N^1-β)ϕ_2 ) error term, giving (<ref>).
 

 Fix ϕ∈ℂ[ c_0, c_1,…, c_0,c_1,…] such that for all ψ∈ℂ[ c_0, c_1,…, c_0,c_1,…] of degree ≤ k we have

    ∫_U(N)ϕ(L_M) ψ(L_M)μ_Haar =0 .



Then for any real-valued ψ∈ℂ[ c_0, c_1,…, c_0,c_1,…] of degree ≤ k, we have

    ∫_U(N)ϕ(L_M)   F( -q^1/2(M),…, -q^k/2 (M^k)/k)/∏_j=1^k  ( e^  -   (M^j)^2 /  j   j q^j / π) μ_Haar
    
    =    ∫_U(N)ϕ(L_M) (  F( -q^1/2(M),…, -q^k/2 (M^k)/k)/∏_j=1^k  ( e^  -   (M^j)^2 /  j   j q^j / π)  - ψ (L_M)  )  μ_Haar + ∫_U (N)ϕ(L_M) ψ(L_M) μ_Haar
    
     =    ∫_U(N)ϕ(L_M) (  F( -q^1/2(M),…, -q^k/2 (M^k)/k)/∏_j=1^k  ( e^  -   (M^j)^2 /  j   j q^j / π)  - ψ (L_M)  )  μ_Haar +0 
    ≤   ϕ_2 √(∫_U(N)(   F( -q^1/2(M),…, -q^k/2 (M^k)/k)/∏_j=1^k  ( e^  -   (M^j)^2 /  j   j q^j / π)  - ψ (L_M) )^2 μ_Haar).


We furthermore have

    ∫_U(N)(  F( -q^1/2(M),…, -q^k/2 (M^k)/k)/∏_j=1^k  ( e^  -   (M^j)^2 /  j   j q^j / π)  - ψ (L_M) )^2 μ_Haar


    =  ∫_(  F( -q^1/2 b_1 ,…, -q^k/2  b_k/√(k) )/∏_j=1^k  ( e^  -  b_j^2   j q^j / π)  - ψ)^2 μ_rm


    = ∫_(  F( -q^1/2b_1,…, -q^k/2b_k /√(k))/∏_j=1^k  ( e^  -    b_k^2  j q^j / π) )^2 μ_rm- 2 ∫_ F( -q^1/2 b_1 ,…, -q^k/2 b_k/√(k) )/∏_j=1^k  ( e^  -b_j^2   j q^j / π) ψμ_rm+ ∫_ψ^2 μ_rm.


We now compare each of these integrals against μ_rm to corresponding integrals against μ_g. First, we have 

    ∫_ψ^2 μ_rm = ∫_ψ^2 μ_g
 by <ref> since ψ^2 has degree 2k ≤ N.

Second, using (<ref>) and then (<ref>) (inputting <ref> and <ref>), we obtain

    ∫_(  F( -q^1/2b_1,…, -q^k/2b_k /√(k))/∏_j=1^k  ( e^  -    b_k^2  j q^j / π) )^2 μ_rm
    
     =   ∫_ℂ^k(  F( -q^1/2b_1,…, -q^k/2b_k /√(k))/∏_j=1^k  ( e^  -    b_k^2  j q^j / π) )^2 η _* μ_rm
    
     =   ∫_ℂ^k(  F( -q^1/2b_1,…, -q^k/2b_k /√(k))/∏_j=1^k  ( e^  -    b_k^2  j q^j / π) )^2 ν_* μ_g  +O ( N^O(1)  e^ -(1- o_N(1)) N^1-βlog(N^1-β ))   
    
     =    ∫_(  F( -q^1/2b_1,…, -q^k/2b_k /√(k))/∏_j=1^k  ( e^  -    b_k^2  j q^j / π) )^2 μ_g+O ( e^ -(1- o_N(1)) N^1-βlog(N^1-β )) .

Third, <ref> gives

    ∫_ F( -q^1/2 b_1 ,…, -q^k/2 b_k/√(k) )/∏_j=1^k  ( e^  -b_j^2   j q^j / π) ψμ_rm
    
    =    ∫_ F( -q^1/2 b_1 ,…, -q^k/2 b_k/√(k) )/∏_j=1^k  ( e^  -b_j^2   j q^j / π) ψμ_g+ O (e^ - (1/2 - o_N(1)) N^1-βlog (N^1-β)ψ_2 ) .
 

Combining (<ref>), (<ref>), and (<ref>), we obtain

    ∫_(  F( -q^1/2 b_1 ,…, -q^k/2  b_k/√(k) )/∏_j=1^k  ( e^  -  b_j^2   j q^j / π)  - ψ)^2 μ_rm


    =  ∫_(  F( -q^1/2 b_1 ,…, -q^k/2  b_k/√(k) )/∏_j=1^k  ( e^  -  b_j^2   j q^j / π)  - ψ)^2 μ_g + O ( e^ -(1- o_N(1)) N^1-βlog(N^1-β )) + O (e^ - (1/2 - o_N(1)) N^1-βlog (N^1-β)ψ_2 ).

To minimize (<ref>), we should choose ψ to be a good approximation in L^2 to F( -q^1/2 b_1 ,…, -q^k/2  b_k/√(k) )/∏_j=1^k  ( e^  -  b_j^2   j q^j / π). To do this we follow Corollary <ref> and set

    ψ =  ∑_ a_1,1, …, a_k,2∈ℤ^≥ 0
    ∑_n=1^k n (a_n,1 + a_n,2) ≤ k  h_a_1,1,…, a_k,2∏_n=1^k  He_a_n,1( x_n,1√(2n/q^n)) He_a_n,2(x_n,2√(2n/q^n) )  √(2n/q^n)^ a_n,1+ a_n,2

so that

    F( -q^1/2 b_1 ,…, -q^k/2  b_k/√(k) )/∏_j=1^k  ( e^  -  b_j^2   j q^j / π)  - ψ
 
    =  ∑_ a_1,1, …, a_k,2∈ℤ^≥ 0
    ∑_n=1^k n (a_n,1 + a_n,2) > k  h_a_1,1,…, a_k,2∏_n=1^k  He_a_n,1( x_n,1√(2n/q^n)) He_a_n,2(x_n,2√(2n/q^n) )  √(2n/q^n)^ a_n,1+ a_n,2

and then

    ∫_(  F( -q^1/2 b_1 ,…, -q^k/2  b_k/√(k) )/∏_j=1^k  ( e^  -  b_j^2   j q^j / π)  - ψ)^2 μ_g


    = ∫_(  ∑_ a_1,1, …, a_k,2∈ℤ^≥ 0
    ∑_n=1^k n (a_n,1 + a_n,2) > k  h_a_1,1,…, a_k,2∏_n=1^k  He_a_n,1( x_n,1√(2n/q^n)) He_a_n,2(x_n,2√(2n/q^n) )  √(2n/q^n)^ a_n,1+ a_n,2)^2 μ_g


    = ∑_ a_1,1, …, a_k,2∈ℤ^≥ 0
    ∑_n=1^k n (a_n,1 + a_n,2) > k h_a_1,1,…, a_k,2^2 ∏_n=1^k a_n,1! a_n,2! ( 2n/q^n)^ a_n,1+ a_n,2≪ k^ - q-2/2≪ N^ - βq-2/2
 by (<ref>) and <ref>.

We also have

    ψ_2^2= ∫_ψ^2 μ_rm = ∫_ψ^2 μ_g
 
    ≤∫_(  F( -q^1/2 b_1 ,…, -q^k/2  b_k/√(k) )/∏_j=1^k  ( e^  -  b_j^2   j q^j / π) )^2  μ_g≤∫_ N^O(1) μ_g = N^O(1).


Plugging these into (<ref>), we obtain

    ∫_(  F( -q^1/2 b_1 ,…, -q^k/2  b_k/√(k) )/∏_j=1^k  ( e^  -  b_j^2   j q^j / π)  - ψ)^2 μ_rm


    = O ( N^ - βq-2/2) + O ( e^ -(1- o_N(1)) N^1-βlog(N^1-β )) + O (e^ - (1/2 - o_N(1)) N^1-βlog (N^1-β)N^O(1) )


    = O ( N^ - βq-2/2)

as the polynomial error term dominates the exponential ones. Plugging this into (<ref>) and using (<ref>) and the fact from <ref> that γ=O(1), this gives (<ref>).




§ REPRESENTATIONS OF THE UNITARY GROUP AND MOMENTS


This section is devoted to the proof of <ref>. We begin in <ref> by describing the relationship between irreducible representations and polynomials in ℂ[c_0,c_1,…, c_0,c_1,…], and between highest weights of representations and the degrees of polynomials. Using this, in <ref> we will prove <ref>, a variant of <ref> with the same error term but a main term expressed very differently in terms of a sum over irreducible representations. The proof of <ref> is very general and should apply with minimal modification beyond moments to other statistics of L-functions such as zero densities and ratios.  The next steps are to give in <ref> an explicit expression, avoiding the language of representation theory, for the main term in <ref>, and to compare the main term of <ref> with the main term of <ref>, leading to a proof of <ref> at the end of this section. It would be possible to avoid the language of representation theory entirely, only writing down explicit polynomials and using the Weyl integration formula, but doing this would make our calculations less motivated.



 §.§ Preliminaries on representations and polynomials


Irreducible representations of the unitary group U(N) are classified by their highest weight, a nonincreasing N-tuple of integers. An irreducible representation of GL_N(ℂ) has highest weight ω_1,…,ω_N if and only if it contains a vector on which upper-triangular unipotent matrices act trivially and on which the diagonal matrix with diagonal entries λ_1,…,λ_N acts by multiplication by ∏_ℓ=1^N λ_ℓ^ω_ℓ. An irreducible representation of U(N) has highest weight ω_1,…,ω_N if and only if it extends to a representation of GL_N(ℂ) with highest weight ω_1,…,ω_N. We denote the highest weight of V by (V). 

We say the norm of the highest weight ω_1,…,ω_N is ∑_ℓ=1^N ω_ℓ. The norm of (V) is denoted by (V).

In this subsection, we will check that polynomials of degree ≤ k in ℂ[c_0,c_1,…, c_0,c_1,…] may be expressed as linear combinations of the characters of irreducible representations with highest weight of norm ≤ k and, conversely, characters of such irreducible representations may be expressed as low-degree polynomials. It follows that characters of irreducible representations with highest weight of norm >k are orthogonal to all low-degree polynomials, an important criterion for applying Theorem <ref>.

 Let V_1 and V_2 be irreducible representations of U(N). Then V_1⊗ V_2 is a sum of irreducible representations of U(N) with highest weights of norms ≤(V_1) + (V_2). 

 If V_1 ⊗ V_2 contains an irreducible summand with highest weight ω_1,…, ω_N then it contains an eigenvector of the diagonal torus with weights ω_1,…, ω_N. Since V_1 and V_2 split as sums of eigenspaces of the diagonal torus, this is only possible if V_1 and V_2 each contain an eigenvector whose weights sum to ω_1,…, ω_N. By linearity of the norm, it suffices to prove that the weights of eigenvectors of V_j have norm at most (V_j). If this were not so, since the set of weights is S_N-invariant, there would have to be a vector whose weight had greater norm with weights in decreasing order, which could not be a sum of the highest weights and negative roots, contradicting the fact that the representation is generated by the highest weight under the negative roots. 

For M ∈ U(N) the definition (<ref>) of L_M implies that 
    L_M (s) = ∑_d=0^n (-1)^n  (M, ∧^d  ) q^d (1/2-s )
 and 
    L_M(s)= ∑_d=0^n (-1)^n  (M, ∧^d ^∨ ) q^d( 1/2  -s).
 

 For ϕ∈ℂ[c_0,c_1,…, c_0,c_1,…], there exists a finite set of irreducible representations V_o and coefficients κ_o such that for all M∈ U(N) we have

    ϕ(L_M) = ∑_o κ_o  (M, V_o)

and if ϕ has degree ≤ d then we can assume that (V_o)≤ d for all o.

 Since all polynomials are linear combinations of monomials, it suffices to prove this for monomials.

We first check for the coefficients of L_M and their complex conjugates. The coefficient of q^-ds is (-q^1/2)^d (M, ∧^d  ) and its complex conjugate is (-q^1/2)^d (M, ∧^d ^∨ ) by (<ref>) and (<ref>). The highest weight of ∧^d has d ones followed by N-d zeroes, while the highest weight of ∧^d ^∨ has N-d zeroes followed by d negative ones, and both of these have norm d.

Any monomial is a product of c_ds and c_ds, hence equal to a constant multiple of a product of traces. The product of traces is the trace of the tensor product, which is the sum of the traces on the irreducible summands of the tensor product. By <ref>, these all have weights with norms bounded by the degree of the monomial. 

  Let ϕ∈ℂ[c_0,c_1,…, c_0,c_1,…] be a polynomial and V an irreducible representation of U(N) such that for all M∈ U(N), 
    ϕ(L_M) =   (M, V) .
 If (V)>k then for all polynomials ψ∈ℂ[c_0,c_1,…, c_0,c_1,…] of degree ≤ k we have
 
    ∫_U(N)ϕ(L_M) ψ(L_M)μ_Haar=0.
 
 
  We apply <ref> to ψ to obtain
 
    ∫_U(N)ϕ(L_M) ψ(L_M)μ_Haar= ∫_U(N)(M,V) ∑_o κ_o  (M, V_o) μ_Haar=∑_o κ_o ∫_U(N)(M,V)  (M, V_o) μ_Haar = 0
 by orthogonality of characters, since V cannot be among the V_o as (V)> k ≥(V_o) for all o. 
 
 Note that the conclusion of <ref> is the assumption (<ref>) of <ref>. To obtain a supply of polynomials to which we can apply <ref>, it suffices to find polynomials ϕ satisfying the hypothesis ϕ(L_M) =  (M, V) of <ref>. We can do this using the Jacobi-Trudi identity for Schur polynomials. 
 
 
 We always take ∧^d  = ∧^d ^∨=0 if d∉[0,N]. For a power series L in q^-s and arbitrary integer d, let c_d(L) be the coefficient of q^-ds in L, so that c_d=0 for d<0.
 

 Let V be a representation of U(N) with highest weight ω_1,…,ω_N. Let  a,b ∈ℤ^≥ 0 be integers satisfying a ≥ω_1 and ω_n ≥ -b. Then




  * (M,V ⊗^b) is the determinant of the a+b × a+b matrix whose ijth entry is (M, ∧^#{ℓ|ω_ℓ≥ i-b } +j-i ).


  * Let d_ij = #{ℓ|ω_ℓ < i-b } + i - j for i≤ b and d_ij =  #{ℓ|ω_ℓ≥ i-b } +j-i for i > b. Then (M,V ) is the determinant of the a+b × a+b matrix whose ijth entry is (M, ∧^ d_ij) for i>b and (M, ∧^d_ij^∨ ) for j>b.


  * (M,V ) is the determinant of the a+b × a+b matrix whose ijth entry is (-q^-1/2)^d_ij c_d_ij (L_M) for i>b and (-q^-1/2)^d_ijc_d_ij (L_M) for i ≤ b.


  * The determinant of the a+b × a+b matrix whose ijth entry is (-q^-1/2)^d_ij c_d_ij for i>b and (-q^-1/2)^d_ij c_d_ij for i ≤ b is a polynomial of degree at most the norm of ω_1,…, ω_n. 

 

 For part (1), if we let _1,…, _N be the eigenvalues of M, then (M,V ⊗^b) is the Schur polynomial in _1,…, _N associated to the partition (ω_1+b,…, ω_N+b) and 
    (M, ∧^#{ℓ|ω_ℓ≥ i-b } +j-i )
 is the #{ℓ|ω_ℓ≥ i-b } +j-ith elementary symmetric polynomial in _1,…, _N. The claim is then a statement of the Jacobi-Trudi identity for Schur polynomials <cit.>.

For part (2), we have (M,V) =  (M,V, ⊗^b) (M)^-b. We take the formula of part (1) and multiply the first b rows by (M)^-1 to multiply the determinant by (M)^-b. This fixes the ij entry for i>b and changes the ij entry for i ≤ b to

    (M, ∧^#{ℓ|ω_ℓ≥ i-b } +j-i )  (M)^-1 =(M, ∧^#{ℓ|ω_ℓ≥ i-b } +j-i ⊗^-1)
 
    =(M, ∧^ N- (#{ℓ|ω_ℓ≥ i-b } +j-i )^∨ ) = (M, ∧^#{ℓ|ω_ℓ < i-b } +i-j)^∨ )  = (M, ∧^d_ij^∨ ).

Part (3) follows from part (2) when we observe that (M, ∧^ d_ij)= (-q^-1/2)^d_ij c_d_ij (L_M) because of (<ref>) and (M, ∧^ d_ij^∨ )= (-q^-1/2)^d_ij c_d_ij (L_M) because of (<ref>).

For part (4), let 
    d(i) = #{ℓ|ω_ℓ <  i-b }   if  i ≤ b 
    #{ℓ|ω_ℓ≥ i-b }   if  i>b
 and let 
    v(i) =   2b -i    if  i ≤ b 
     i     if  i>b .
 Then if i,j> b we have v(j)-v(i)=j-i, if j≤ b < i we have v(j)-v(i) = 2b-j -i  ≥ j-i, if i,j ≤ b we have v(j)-v(i) = i-j, and if i ≤ b < j we have v(j)-v(i)=  j + i-2b  ≥  i-j so in all cases we have 
    d(i) + v(j)-v(i) ≥ d_ij .
 The ij-entry is a polynomial of degree d_ij. This implies the determinant has degree ≤∑_i=1^a+b d(i) since when we calculate the degree of each term in the Leibniz expansion, the v(j) and v(i) cancel. Finally ∑_i=1^a+b d(i) is the norm of ω_1,…, ω_N since if ω_ℓ≥ 0 then ω_ℓ contributes to d(b+1),…, d(b+ ω_ℓ) and thus contributes ω_ℓ to the sum while if ω_ℓ≤ 0 then ω_ℓ contributes to d(b+1+ω_ℓ),…, d(b) and thus contributes -ω_ℓ to the sum. 



 §.§ Handling the error term

 
 
  Fix ,, and N. There exist coefficients κ_o (depending on s_1,…, s_+) and irreducible representations V_o such that for each M∈ U(N) we have
 
    ∏_i=1^ L_M(s_i ) ∏_i=+1^ + L _M( s_i ) =  ∑_o κ_o (M, V_o).

  By (<ref>) and (<ref>), L_M(s) and L_M(s) may be expressed as complex-linear combinations of characters of U(N). Multiplying these expressions, it follows that 
∏_i=1^ L_M(s_i ) ∏_i=+1^ + L _M( s_i ) is a complex-linear combinations of characters of U(N). 



Using <ref>(3), we associate to each V_o a polynomial ψ_o∈ℂ[c_0, c_1,…, c_0,c_1,…] such that ψ_o (L_M) = (M, V_o). This gives 

    ∏_i=1^ L(s_i ) ∏_i=+1^ + L ( s_i ) =  ∑_o κ_o ψ_o(L)
 as long as L= L_M for some M∈ U(N).

We define 
    ϕ_lf = ∑_ o ,   (V_o)≤ kκ_o ψ_o
 and 
    ϕ_hf = ∑_ o ,   (V_o)> kκ_oψ_o.
  

Note that all implicit constants in big O notation used in this section will be allowed to depend on ,,q but not on N (since the final goal is to prove <ref>, an estimate whose error term depends on ,,q but not on N).


 Both ϕ_hf_2 and ϕ_lf_2 are O( N ^(+)^2 /2).
 We have 
    ϕ_hf + ϕ_lf^2 = ∫_U(N)ϕ_hf(L_M)  +  ϕ_lf(L_M) ^2 μ_Haar =  ∫_U(N)∏_i=1^ L_M(s_i ) ∏_i=+1^ + L _M( s_i ) ^2  μ_Haar
 
    = ∏_i=1^+( ∫_U(N)L_M(s_i)^2(+)μ_Haar)^1/(+) =    ∫_U(N)L_M(1/2)^2(+)μ_Haar = O( N ^ (+)^2 )

by Hölder's inequality, the invariance under translation by diagonal matrices of Haar measure on U(N), and the classical calculation of the moments of the characteristic polynomial of random unitary matrices. By <ref> and <ref>(4) we have  
    ∫_U(N)ϕ_hf(L_M) ϕ_lf(L_M)μ_Haar=0,
 i.e. ϕ_lf and ϕ_hf are orthogonal, so 
ϕ_hf_2^2+ ϕ_lf_2^2 = ϕ_hf + ϕ_lf^2 and thus the indvidual norms are bounded as well.



 Assume that q>11. Let  and  be nonnegative integers and s_1,…, s_ + be complex numbers with real part 1/2. We have 
    ∫_( ∏_i=1^ L(s_i ) ∏_i=+1^ + L ( s_i ) ) μ_ch = ∫_ℂ[[q^-s ]]ϕ_lfμ_ep + O ( N^(+)^2/2  -βq-2/4).
 

 From (<ref>) and the definitions of ϕ_lf and ϕ_hf we have 
    ∫_( ∏_i=1^ L(s_i ) ∏_i=+1^ + L ( s_i ) ) μ_ch = ∫_∑_o κ_o ψ_o μ_ch
     =    ∫_ℂ[[q^-s ]]  (ϕ_lf +ϕ_hf ) μ_ch =∫_ℂ[[q^-s ]]ϕ_lfμ_ch + ∫_ℂ[[q^-s ]]ϕ_hfμ_ch.
 

To ∫_ℂ[[q^-s ]]ϕ_lfμ_ch we apply <ref>, using <ref>(4) to check the hypothesis, and to ∫_ℂ[[q^-s ]]ϕ_hfμ_ch we apply <ref>, using <ref> to check the hypothesis (<ref>). From these results and (<ref>) we obtain

    ∫_( ∏_i=1^ L(s_i ) ∏_i=+1^ + L ( s_i ) ) μ_ch = ∫_ℂ[[q^-s ]]ϕ_lfμ_ep + O (e^ (1/2 - o_N(1)) N^1-βlog (N^1-β)ϕ_lf_2 )+ O ( N^-  βq-2/4ϕ_hf_2   ) .


  Since e^ (1/2 - o_N(1)) N^1-βlog (N^1-β) is bounded by N^-  βq-2/4, <ref> together with (<ref>) gives (<ref>). 



 §.§ Comparing the main terms


To prove <ref>, it remains to compare ∫_ℂ[[q^-s ]]ϕ_lfμ_ep to MT^,_N(s_1,…, s_+).

To begin, we will calculate ϕ_lf more precisely, which requires making the representations V_o and coefficients κ_o appearing in (<ref>) explicit. We also make use of the change of variables α_i = s_i- 1/2, so in particular q^1/2-s_i= q^-α_i and α_i is imaginary so α_i=-α_i.

Our calculation will culminate in the formula (<ref>) which expresses ∫_ℂ[[q^-s ]]ϕ_lfμ_ep using a sum over polynomials ψ_𝐞 against coefficients κ_𝐞 indexed by certain tuples of integers 𝐞. To motivate the definitions of κ_𝐞 and ψ_𝐞 the proof will proceed in steps.

After proving (<ref>), we will equate a certain longer sum to MT^,_N. The difference between this longer sum and the original introduces a secondary error term which we will also bound.

To prove (<ref>), we use the method of Bump and Gamburd <cit.>, i.e. we apply the Cauchy identity for Schur functions to express the desired moment as a sum of products of pairs of Schur functions. One Schur function in each pair will beocme κ_𝐞 and the other will become ψ_𝐞. This method was originally used to calculate expectations of products of the characteristic polynomial of a unitary matrix against Haar measure, but here we apply it (together with other tools) to calculate the expectation againt a non-uniform measure.




If _1(M),…, _N(M) are the eigenvalues of M then 
    L_M(s_i) = ( I - q^ -α_i  M ) = ∏_ℓ=1^N  ( 1- q^-α_i _ℓ(M))
 while 
    L_M(s_i) = ( I - q^ -α_i M )  =(-1)^N q^ N α_i ( M)^-1( I - q^α_i   M^-1 )
 
    = (-1)^N q^ N α_i  ( M)^-1( I - q^-α_i  M ) = (-1)^N q^ N α_i ( M)^-1∏_ℓ=1^N ( 1 -q^-α_i _N(M))


so 
    ∏_i=1^ L_M(s_i ) ∏_i=+1^ + L _M( s_i )  = (-1)^N  ( M)^-∏_i=+1^+ q^   N α_i  ∏_i=1^+∏_ℓ=1^N  ( 1 -q^-α_i _N(M)) .


The Cauchy identity for Schur functions gives 
    ∏_i=1^+∏_ℓ=1^N  ( 1 -q^-α_i _N(M)) = ∑_ρ s_ρ ( q^-α_1,…, q^-α_+ ) s_ρ' ( _1(M),…, _N(M))
 where ρ denotes a partition, s_ρ the Schur function associated to that partition, ρ' the dual partition, and s_ρ' the corresponding Schur function. The +-variable Schur function s_ρ vanishes unless ρ has at most + parts and the N-variable Schur function s_ρ' vanishes unless all parts of ρ have size at most ≤ N. Partitions satisfying both of these can be equivalently expressed as tuples e_1,…, e_+ of integers satisfying N ≥ e_1 ≥…≥ e_+≥ 0, giving

    ∏_i=1^+∏_ℓ=1^N  ( 1 -q^α_i_N(M)) =∑_ e_1,… e_+∈ℤ
     N ≥ e_1 ≥…≥ e_+≥ 0  s_(e_1,…, e_+) (  q^-α_1,…, q^-α_+ )  s_ (e_1,…, e_+)' ( _1(M),…, _N(M) )

and thus

    ∏_i=1^ L_M(s_i ) ∏_i=+1^ + L _M( s_i )
 
    = ∑_ e_1,… e_+∈ℤ
     N ≥ e_1 ≥…≥ e_+≥ 0 ( (-1)^N ∏_i=+1^+ q^   Nα_i   s_(e_1,…, e_+) (  q^-α_1,…, q^-α_+ ) ) ( ( M)^-  s_ (e_1,…, e_+)' ( _1(M),…, _N(M) )) .

Now the significance of this expression is that s_ (e_1,…, e_+)' ( _1(M),…, _N(M) ) is the trace of M acting on the irreducible representation of U(N) with highest weight (e_1,…, e_+)' so that ( M)^-  s_ (e_1,…, e_+)' ( _1(M),…, _N(M) ) is the trace of M acting on the irreducible representation of U(N) with highest weight obtained from (e_1,…, e_+)' by subtracting  from each entry. We refer to this representation as V_𝐞. These representations V_𝐞 have distinct highest weights, and thus are not isomorphic, for distinct V_𝐞. Thus the expression

    ∏_i=1^ L_M(s_i ) ∏_i=+1^ + L _M( s_i )  = ∑_ e_1,… e_+∈ℤ
     N ≥ e_1 ≥…≥ e_+≥ 0 ( (-1)^N ∏_i=+1^+ q^   Nα_i   s_(e_1,…, e_+) ( q^-α_1,…, q^-α_+) ) (M, V_𝐞)

is a precise form of (<ref>). 

We now let ψ_𝐞 be the determinant of the +× + matrix whose ijth entry is (-q^-1/2)^  e_i + j-i c_ e_i+j-i for i> and (-q^-1/2)^ N-e_i + i - j c_ N-e_i +i -j for i ≤. 

 


  * ψ_𝐞 is the polynomial associated to V_𝐞 by <ref>(3) with a= and b=.


  * The norm of the highest weight of V_𝐞 is ∑_i=1^ (N-e_i)  + ∑_i=+1^+ e_i.  

 (e_1,…, e_+)' is the vector consisting of e_+ copies of +,  e_i-e_i+1 copies of i for all i from +-1 to 1, and N-e_1 copies of 0. Subtracting  from each entry gives the vector ω_1,…, ω_N with e_+ copies of ,  e_i-e_i+1 copies of i- for all i from +-1 to 1, and N-e_1 copies of -.

It follows that #{ℓ|ω_ℓ < i-} is N- e_i and #{ℓ|ω_ℓ≥ i-} is e_i. With notation as in <ref>(2), this implies d_ij= e_i +j-i for i> and d_ij= N-e_i+i-j for i≤, which plugged into <ref>(3) gives ψ_𝐞.

The norm of ω_1,…,ω_N is   
    e_+r  + ∑_i=1^+-1 (e_i -e_i+1 )   i - + (N-e_1)  =  ∑_i=1^+ e_i  (i- - i--1)  + N  = ∑_i=1^ (N-e_i) + ∑_i=+1^+ e_i
 by a telescoping sum. 

We also let  
    κ_𝐞 =  (-1)^N ∑_σ∈ S_+(σ) ∏_i=1^+ q^ - (e_i ++- i)  α_σ(i) /∏_1 ≤ i_1 < i_2 ≤+ (q^-α_i_1 - q^ - α_i_2 ) ∏_i=+1^+ q^   Nα_i    .


That  
    κ_𝐞 = (-1)^N   s_(e_1,…, e_+) ( q^-α_1,…, q^-α_+)  ∏_i=+1^+q^   Nα_i
 follows from the definition of Schur polynomials (or, defining them in terms of irreducible representations, from the Weyl character formula).


(<ref>), <ref>, and (<ref>) imply that 
    ϕ_lf =  ∑_ e_1,… e_+∈ℤ
     N ≥ e_1 ≥…≥ e_+≥ 0 
    ∑_i=1^ (N - e_i) + ∑_i=+1^+ e_i ≤ k  κ_𝐞ψ_𝐞

and therefore

    ∫_ϕ_lfμ_ep =  ∑_ e_1,… e_+∈ℤ
     N ≥ e_1 ≥…≥ e_+≥ 0 
    ∑_i=1^ (N - e_i) + ∑_i=+1^+ e_i ≤ k  κ_𝐞∫_ψ_𝐞μ_ep.


However, we have defined both κ_𝐞 and ϕ_𝐞 to make sense for an arbitrary tuple of integers 𝐞, not the ones where the Schur polynomials are defined. We will use this flexibility to extend the range of summation, which will allow us to compare this sum to MT_N^,, which is similarly formed by extending a sum beyond the obviously appropriate range. 

Observe first that (<ref>) implies that

    ∫_ϕ_lfμ_ep =  ∑_ e_1,… e_+∈ℤ
     N ≥ e_1≥… e_
     e_+1≥…  e_+≥ 0 
    ∑_i=1^ (N - e_i) + ∑_i=+1^+ e_i ≤ k  κ_𝐞∫_ψ_𝐞μ_ep

since the sole condition appearing in (<ref>) but not (<ref>) is e_≥ e_+1, but this is implied by the other conditions of (<ref>) since  e_ - e_+1 = N - (N-e_) - e_+1≥ N -k ≥ 0.

Our next two goals will be to check that

    ∑_ e_1,… e_+∈ℤ
     N ≥ e_1≥… e_
     e_+1≥…  e_+≥ 0   κ_𝐞∫_ψ_𝐞μ_ep =  MT_N^,

and

    ∑_ e_1,… e_+∈ℤ
     N ≥ e_1≥… e_
     e_+1≥…  e_+≥ 0 
    ∑_i=1^ (N - e_i) + ∑_i=+1^+ e_i > k  κ_𝐞∫_ψ_𝐞μ_ep =  O (k^ O(1) q^ - k/4 ).


For both of these, we begin by evaluating ∫_ψ_𝐞μ_ep explicitly in terms of polynomials in 𝔽_q[u].  This evaluation in Lemma <ref> will let us prove a bound for ∫_ψ_𝐞μ_ep in Lemma <ref>. This bound will be used to prove (<ref>) and then both the evaluation and the bound will be used to prove (<ref>). 

 For L= ∑_d=0^∞ c_d q^-d s, we have an identity of formal Laurent series in q^-α_1,… ,q^-α_+

    ∑_e_1,…, e_+∈ℤψ_𝐞∏_i=1^ (-q^α_i)^-e_i∏_i=+1^+ (-q^-α_i)^e_i
  
    = ∏_1 ≤ i_1< i_2 ≤+ (q^α_i_1 - q^α_i_2)  ∏_i=1^( (-q^α_i)^1-i-NL (q^- 1/2 - α_i)) ∏_i=+1^+  ((-q^-α_i)^i-1L ( q^- 1/2 - α_i )  ).
 

 Consider the +×+ matrix whose ijth entry is 

    ∑_e_i ∈ℤ (-q^-1/2)^  e_i + j-i c_ e_i+j-i(L)  (- q^-α_i)^e_i =  (-q^-α_i)^i-j  L ( q^- 1/2 - α_i )
 for i> and 

    ∑_e_i ∈ℤ (-q^-1/2)^ N-e_i + i - j c_ N-e_i +i -j    (-q^α_i)^- e_i =(-q^α_i)^j-i-NL (q^- 1/2 - α_i)

for i ≤. 

By additivity of determinants in each row, the determinant of this matrix is (<ref>).

On the other hand, removing a factor of (-q^-α_i)^i-1L ( q^- 1/2 - α_i ) from the ith row for i> and (-q^α_i)^1-i-NL (q^- 1/2 - α_i) from the i'th row for i≤, we obtain the matrix whose ij-entry is ( -q^α_i)^j-1 for all i,j, which is a Vandermonde matrix and thus has determinant 

    ∏_1 ≤ i_1< i_2 ≤+  ( - q^α_i_2 -  (- q^α_i-1)) = ∏_1 ≤ i_1< i_2 ≤+ (q^α_i_1 - q^α_i_2)

so by the compatibility of determinants with scalar multiplication of rows, the determinant is also (<ref>). 

 We have an identity of formal Laurent series in q^-α_1,… ,q^-α_+

    ∫_∏_i=1^L (q^- 1/2 - α_i) ∏_i=+1^+ L ( q^- 1/2 - α_i ) μ_ep  = ∑_f_1,…, f_+∈𝔽_q[u]^+ 
    ∏_i=1^ f_i =∏_i=+1^+ f_i ∏_i=1^f_i^-1/2+α_i∏_i=+1^+f_i^-1/2-α_i
 where the integration is applied separately to each term in the formal Laurent series.  

 Expanding out we have 

    ∏_i=1^L_ξ (q^- 1/2 + α_i) ∏_i=+1^+ L_ξ ( q^- 1/2 - α_i ) = ∑_f_1,…, f_+∈𝔽_q[u]^+ ∏_i=1^(ξ(f_i)f_i^-1/2+α_i  )∏_i=+1^+( ξ(f_i) f_i^-1/2-α_i)

and orthogonality of characters on a product of circle groups gives

    𝔼[  ∏_i=1^ξ(f_i)∏_i=+1^+ξ(f_i)] =  1    if ∏_i=1^ f_i =∏_i=+1^+ f_i  
     0    otherwise.

Taking the expectation of (<ref>) over ξ and then plugging in (<ref>) gives (<ref>). 


 For each e_1,…, e_+∈ℤ, ∫_ψ_𝐞μ_ep is (-1)^∑_i=1^+ e_i + +2 +N times the coefficient of q^∑_i=1^(i-1+N-e_i) α_i - ∑_i=+1^+ (d_i-i+1)α_i in 
    ∏_1 ≤ i_1< i_2 ≤+ (q^α_i_1 - q^α_i_2)   ∑_f_1,…, f_+∈𝔽_q[u]^+ 
    ∏_i=1^ f_i =∏_i=+1^+ f_i ∏_i=1^f_i^-1/2+α_i∏_i=+1^+f_i^-1/2-α_i.
  

 Integrating Lemma <ref> against μ_ep and then plugging in (<ref>) gives

    ∑_e_1,…, e_+∈ℤ∫_ψ_𝐞μ_ep∏_i=1^ (-q^α_i)^-e_i∏_i=+1^+ (-q^-α_i)^e_i
  
    = ∏_1 ≤ i_1< i_2 ≤+ (q^α_i_1 - q^α_i_2)  ∏_i=1^ (-q^α_i)^1-i-N∏_i=+1^+  (-q^-α_i)^i-1∑_f_1,…, f_+∈𝔽_q[u]^+ 
    ∏_i=1^ f_i =∏_i=+1^+ f_i ∏_i=1^f_i^-1/2+α_i∏_i=+1^+f_i^-1/2-α_i .
 

Moving some factors to the left-hand side, we obtain

    ∑_e_1,…, e_+∈ℤ∫_ψ_𝐞μ_ep∏_i=1^ (-q^α_i)^i-1+N-e_i∏_i=+1^+ (-q^-α_i)^e_i-i+1
 
    = ∏_1 ≤ i_1< i_2 ≤+ (q^α_i_1 - q^α_i_2)   ∑_f_1,…, f_+∈𝔽_q[u]^+ 
    ∏_i=1^ f_i =∏_i=+1^+ f_i ∏_i=1^f_i^-1/2+α_i∏_i=+1^+f_i^-1/2-α_i .
 
Extracting the coefficient of a single term and grouping together all the powers of (-1), we obtain the statement. 

 We have   
    ∫_ψ_𝐞μ_ep = O (  (O(1) + ∑_i=1^ (N-e_i) + ∑_i=+1^+ e_i)^O(1)  q^- max ( ∑_i=1^ (N-e_i), ∑_i=+1^+ e_i /2)).
 Furthermore, ∫_ψ_𝐞μ_ep =0 unless ∑_i=1^(i-1+N-e_i)  - ∑_i=+1^+ (e_i-i+1) = +2.   

 We will apply bounds in <cit.> for the coefficients of the series M_S, defined in <cit.> as
 
    M_S (α_1,…,α_+) =  ∏_1 ≤ i_1< i_2 ≤+ (q^α_i_1 - q^α_i_2) ∑_ f_1,…, f_+∈𝔽_q[u]^+ 
    ∏_i ∈ S f_i  /  ∏_i∉ S f_i ∈ u^ℤ∏_i ∈ Sf_i ^ -1/2 + α_i∏_i∉ Sf_i ^-1/2 - α_i   .
 
 Taking S = {1,…,}, this definition specializes to
 
    M_{1,…,}(α_1,…,α_+) =  ∏_1 ≤ i_1< i_2 ≤+ (q^α_i_1 - q^α_i_2) ∑_ f_1,…, f_+∈𝔽_q[u]^+ 
    ∏_i =1^  f_i  /  ∏_i=+1^+  f_i ∈ u^ℤ∏_i=1^f_i^-1/2+α_i∏_i=+1^+f_i^-1/2-α_i .

(<ref>) and (<ref>) agree except that the condition ∏_i =1^  f_i  /  ∏_i=+1^+  f_i ∈ u^ℤ in (<ref>) is laxer than the condition ∏_i =1^  f_i  =  ∏_i=+1^+  f_i in (<ref>).

If f_1,…, f_+ satisfy ∏_i =1^  f_i  /  ∏_i=+1^+  f_i = u^n for n∈ℤ, then ∑_i=1^ f_i = ∑_i=+1^+ f_i +n, so ∏_i=1^f_i^-1/2+α_i∏_i=+1^+f_i^-1/2-α_i is a monomial in q^α_1,…, q^α_+ of total degree n. Since the Vandermonde ∏_1 ≤ i_1< i_2 ≤+ (q^α_i_1 - q^α_i_2) has total degree +2 in q^α_1,…, q^α_+, this implies that (f_1,…, f_+) contributes to terms in the formal Laurent series (<ref>) with total degree n + +2. Thus, restricting to f_1,…, f_+ satisfying ∏_i =1^  f_i  =  ∏_i=+1^+  f_i, i.e., restricting to the case n=0, is equivalent to restricting the series (<ref>) to terms of total degree +2.

In particular, since ∫_ψ_𝐞μ_ep is by <ref> ± the coefficient of q^∑_i=1^(i-1+N-e_i)α_i - ∑_i=+1^+ (d_i- i+ 1)α_i in (<ref>), it follows that ∫_ψ_𝐞μ_ep is either ± the coefficient of q^∑_i=1^(i-1+N-e_i)α_i - ∑_i=+1^+ (e_i-i+1 )α_i in (<ref>) or equal to zero.

 Hence any upper bound on the coefficients of (<ref>) also gives an upper bound on  ∫_ψ_𝐞μ_ep, which we will shortly use to establish the first part of the statement.  Furthermore, the case where ∫_ψ_𝐞μ_ep =0 occurs when the total degree ∑_i=1^(i-1+N-e_i)  - ∑_i=+1^+ (e_i-i+1) of  q^∑_i=1^(i-1+N-e_i)α_i - ∑_i=+1^+ (e_i-i+1 )α_i is not equal to +2, giving the second part of the statement.

We apply the upper bound <cit.>, which is stated as an upper bound on the coefficient of ∏_i=1^+ q^α_i d_i, so we must substitute in ( N-e_1  ,…, N- e_ + -1 , - e_+1+, …, -e_+ + + -1) for (d_1,…,d_+) in the bounds of <cit.>. (Also, d_1,…, d_+ are required to satisfy the inequalities of <cit.>, but <cit.> guarantees the coefficient vanishes if the inequality is not satisfied, so the bound holds also in that case.)  The bound of <cit.>, is a product of two factors, the first of which is 
    O (  ( O(1) + ∑_i ∈ S d_i - ∑_i∉ S d_i)^O(1) )   = O (  ( O(1) + ∑_i =1^  d_i - ∑_i=+1^+ d_i)^O(1) )
 and substituting gives 
    O ((O(1) + ∑_i=1^ (N-e_i) + ∑_i=+1^+ e_i)^O(1))
 since the i-1 terms may be absorbed into the O(1). The second factor is the minimum of four different bounds, of which we will only need the middle two, which are

    min( q ^ -∑_i∈ S d_i +S2/2, q^∑_i∉ S d_i   + S2 - +2/2) =  min( q ^-  -∑_i=1^  d_i + 2/2, q^∑_i=+1^+ d_i   + 2 - +2/2)

and substituting gives

    min( q ^ -∑_i=1^  (N-e_i )/2, q^ - ∑_i=+1^+ e_i /2) =q^ - max ( ∑_i=1^ (N-e_i), ∑_i=+1^+ e_i )/2
 since the - ∑_i=1^ (i-1) term cancels 2 in the exponent of the first q and ∑_i=+1^+  (i-1) cancels  2 - +2 in the exponent of the second q. 

  We have 
    ∑_ e_1,… e_+∈ℤ
     N ≥ e_1≥… e_
     e_+1≥…  e_+≥ 0 
    ∑_i=1^ (N - e_i) + ∑_i=+1^+ e_i > k  κ_𝐞∫_ψ_𝐞μ_ep =  O ( N^ O(1) q^ - k/4 ).
 

We have

    κ_𝐞 =  ∑_σ∈ S_+(σ) ∏_i=1^+ q^ - (e_i +  + - i)  α_σ(i) /∏_1 ≤ i_1 < i_2 ≤+ (q^-α_i_1 - q^ - α_i_2 ) ≤∏_ 1≤ i_1 < i_2 ≤ + e_i_1 -i_1 - e_i_2 + i_2/∏_ 1≤ i_1 < i_2 ≤ +i_1- i_2
 since the value of the Weyl character formula for the trace of the unitary representation at a unitary matrix with eigenvalues q^ -α_1,… ,q^-α_+ is bounded by the dimension of that representation which is given by the Weyl dimension formula. (Even if e_1 ≥ e_2 ≥… e_+ is not satisfied, the Weyl character formula still gives a formula for the either plus or minus the trace of some irreducible representation, or the zero representation, and the Weyl dimension formula gives plus or minus the dimension of that representation so the absolute value of the dimension formula still bounds the absolute value of the character formula.) 

If N ≥ e_1≥…≥ e_ and e_+1≥…  e_+≥ 0 then each factor e_i_1 -i_1 - e_i_2 + i_2 is certainly bounded by ∑_i=1^ (N-e_i) + ∑_i=+1^+ e_i + O(N) so 
    κ_𝐞≤(O (N) + ∑_i=1^ (N-e_i) + ∑_i=+1^+ e_i)^O(1)
 which together with <ref> implies 

    κ_𝐞∫_ψ_𝐞μ_ep = O (  (O (N) + ∑_i=1^ (N-e_i) + ∑_i=+1^+ e_i)^O(1)   q^- max ( ∑_i=1^ (N-e_i), ∑_i=+1^+ e_i )/2)
 
    =  O (  (O (N) + ∑_i=1^ (N-e_i) + ∑_i=+1^+ e_i)^O(1)   q^- ∑_i=1^ (N-e_i)-∑_i=+1^+ e_i /4) .


The number of tuples e_1,…, e_+ satisfying N ≥ e_1≥… e_ and e_+1≥…  e_+≥ 0 and ∑_i=1^ (N-e_i) + ∑_i=+1^+ e_i = d is O ( d^O(1) ), so in total we have

    ∑_ e_1,… e_+∈ℤ
     N ≥ e_1≥… e_
     e_+1≥…  e_+≥ 0 
    ∑_i=1^ (N - e_i) + ∑_i=+1^+ e_i > k  κ_𝐞∫_ψ_𝐞μ_ep= ∑_d=k+1^∞  O ( d^O(1) ) O ( (O(N)+ d)^ O(1)  q^ - d/4 )
 
    ≪ N^O(1)∑_d=k+1^∞ d^O(1) q^- d/4 = O ( N^O(1) k^O(1) q^ - k/4 ) = O ( N^ O(1) q^ - k/4 ).
 


Recall that our desired main term has the form (after substituting 1/2+α_i for s_i)

    MT_N^, (1/2+α_1,…, 1/2+α_+ )= ∏_i=+1^+ q^α_i N∑_  S ⊆{1,…,+}
    S=∏_i ∈ S q^ -α_i N∑_ f_1,…, f_+∈𝔽_q[u]^+ 
    ∏_i ∈ S f_i = ∏_i∉ S f_i∏_i∈ Sf_i ^ -1/2- α_i  ∏_i ∉ Sf_i ^ -1/2-α_i   .

This is interpreted by continuing each term ∑_ f_1,…, f_+∈𝔽_q[u]^+ 
∏_i ∈ S f_i = ∏_i∉ S f_i∏_i∈ Sf_i ^ -1/2-α_i  ∏_i ∉ Sf_i ^ -1/2+α_i meromorphically from its domain of absolute convergence and then summing the meromorphic functions. Thus, in this segment of the proof only, we will allow the α_i to be arbitrary complex numbers instead of imaginary numbers. We first evaluate the summand associated to S ={1,…, }, before using this to evaluate the summand associated to arbitrary S, and finally evaluate the full main term.


 We have an equality of holomorphic functions on  
    {α_1,…, α_+∈ℂ|Re(α_i)< 1/4 for  i ≤, Re(α_i)> - 1/4 for  i > }
 given by 
    ∏_1 ≤ i_1 < i_2 ≤+ (q^-α_i_1 - q^ - α_i_2 )  ∏_i=1^ q^ - α_iN ∑_ f_1,…, f_+∈𝔽_q[u]^+ 
    ∏_i=1^ f_i = ∏_i=+1^+ f_i∏_i=1^f_i ^ -1/2+α_i  ∏_i=+1^+f_i ^ -1/2-α_i

    =  ∑_ e_1,… e_+∈ℤ
     N ≥ e_1≥… e_
     e_+1≥…  e_+≥ 0   ∑_σ∈ S_× S_(σ) ∏_i=1^+ q^ - (e_i ++- i)  α_σ(i) ∫_ψ_𝐞μ_ep
 where (<ref>) is interpreted by holomorphic continuation from its domain of absolute convergence and (<ref>) is absolutely convergent, and where S_× S_ is embedded in S_ + as the subgroup preserving the partition into {1,…, } and {+1,…, +}. 

 We first check absolute convergence of (<ref>).   By <ref> we have

    ∫_ψ_𝐞μ_ep = O (  (O(1) + ∑_i=1^ (N-d_i) + ∑_i=+1^+ d_i)^O(1)  q^- ∑_i=1^ (N-d_i) - ∑_i=+1^+ d_i /4 ).

For i ≤ we have  
    q^ - (e_i ++- i)  α_σ(i)  = q^ - (e_i ++- i)  Re (α_σ(i) )
 
    = q^  (N-e_i) Re (α_σ(i) )  q^ - (N + +-i) Re (α_σ(i) )≪_N, α q^  (N-e_i) Re (α_σ(i) ) ≤  q^ (N-e_i) max_1≤ j ≤Re (α_j )
 where ≪_N,α denotes an implicit constant that may depend on N and α_1,…, α_+ but does not depend on e_1,…, e_+ (which is all that is needed for absolute convergence). Similarly, for i> we have 
    q^ - (e_i ++- i)  α_σ(i)  = q^ - (e_i ++- i)  Re (α_σ(i) )  = q^ -e_i  Re (α_σ(i) )   q^ - (+-i) Re (α_σ(i) )
 
    ≪_α q^ -e_i  Re (α_σ(i) ) ≤ q^ -e_i min_+1 ≤ j ≤+Re (α_j )

so

    ∏_i=1^+ q^ - (e_i ++- i)  α_σ(i) ≪_N,α   q^∑_i=1^ (N-e_i) max_1≤ j ≤Re (α_j )   - ∑_i=+1^+ e_i min_+1 ≤ j ≤+Re (α_j  )

and thus

    ∏_i=1^+ q^ - (e_i ++- i)  α_σ(i) ∫_ψ_𝐞μ_ep
 
    ≪_N,α  q^ -∑_i=1^ (N-e_i) ( 1/4 - max_1≤ j ≤Re (α_j ) )   - ∑_i=+1^+ e_i ( 1/4+ min_+1 ≤ j ≤+Re (α_j  )  )       O (  (O(1) + ∑_i=1^ (N-e_i) + ∑_i=+1^+ e_i)^O(1))

When we sum over e_1,…, e_+, the assumptions on Re(α_i) imply that the exponential term in q is exponentially decreasing and thus dominates the polynomial term and leads to absolute convergence of (<ref>).

Both (<ref>) and (<ref>) may be expressed as formal Laurent series in q^-α_1, …, q^-α_+. It now suffices to check that, for each d_1,…, d_+∈ℤ, the coefficient of q^∑_i=1^+ d_i α_i in  (<ref>) equals the coefficient of q^∑_i=1^+ d_i α_i in  (<ref>). Then both sides will be equal on the locus where both are absolutely convergent, hence equal everywhere by analytic continuation. This in particular implies (<ref>) is holomorphic on the same region as (<ref>).

To check the equality of coefficients, we first observe that ∑_σ∈ S_× S_(σ) ∏_i=1^+ q^ - (e_i ++- i)  α_σ(i) is antisymmetric in the variables α_1,…,α_ in the sense that swapping two variables is equivalent to multiplying the sum by -1, and similarly antisymmetric in the variables α_+1,…, α_+. Antisymmetry is stable under linear combinations, so (<ref>) is antisymmetric in α_1,…,α_ and in α_+1,…, α_+. Similarly, ∑_ f_1,…, f_+∈𝔽_q[u]^+ 
∏_i=1^ f_i = ∏_i=+1^+ f_i∏_i=1^f_i ^ -1/2-α_i  ∏_i=+1^+f_i ^ -1/2+α_i is symmetric in  α_1,…,α_ and in α_+1,…, α_+ and the Vandermonde is antisymmetric, so their product (<ref>) is antisymmetric.

It follows that the coefficient of q^∑_i=1^+ d_i α_i in either (<ref>) or (<ref>) is antisymmetric in d_1,…, d_ and in d_+1,…, d_+. In particular, the coefficient vanishes if we have d_i=d_j for i< j ≤ or +1< i<j, as in that case swapping d_i and d_j both preserves the value of the coefficient and negates it.  Furthermore, to check that the coefficients of q^∑_i=1^+ d_i α_i are equal for all d_1,…, d_+, it suffices to check for only tuples such that d_1< … < d_ and d_+1< … < d_+: If the d_1,…, d_ are distinct, we can swap them until they are in increasing order, multiplying both coefficients by the same power of -1, and similarly with the d_+1,…, d_+, but if they are not distinct, both coefficients vanish and are trivially equal.

So it remains to check for d_1<… < d_ and d_+1< … <d_+ that the coefficients of q^∑_i=1^+ d_i α_i in  (<ref>) and  (<ref>) are equal. First, we observe that the only σ that contributes to this coefficient in (<ref>) is the identity, since we have e_1 ≥…≥ e_ and e_+1≥… e_+ so that -(e_1++-1 )<-( e_2 + +-2) < … <-( e_+) and  -(e_+1 + -1)< … <- e_+ and any permutation other than the identity would change the order and thus not take these exponents to d_1,… , d_+.  So the only relevant term is the one with σ = id and e_i =  i - - -d_i for all i. 

If N ≥ 1-- -d_1 and d_+≤ 0  so that N ≥ e_1 and e_+≥ 0 then this  term has coefficient ∫_ψ_𝐞μ_ep which by <ref> is 
    (-1)^∑_i=1^+ e_i + +2 +N =(-1)^∑_i=1^+ d_i+N
 times the coefficient of q^∑_i=1^(+ -1+N +d_i ) α_i - ∑_i=+1^+ (- d_i - -   +1)α_i in (<ref>).

Even if N < 1--   -d_1 or d_+ > 0 then the same conclusion holds as then no term in (<ref>) contributes so the coefficient is zero but it is compared to the coefficient in (<ref>) of a monomial whose exponent in q^α_1 is <- N--  +1 or whose exponent of q^α_+ is > 0 and no monomial of this form appears so this is also trivially zero.

On the other hand, (<ref>) is equal to the product of (<ref>) with 
    ∏_1 ≤ i_1 < i_2 ≤+ (q^-α_i_1 - q^ - α_i_2 ) /∏_1 ≤ i_1 < i_2 ≤+ (q^α_i_1 - q^α_i_2 )∏_i=1^  q^ - α_iN    =   ∏_1 ≤ i_1 < i_2 ≤+( - q^-α_i_1 q^- α_i_2) ∏_i=1^ q^ - α_iN
 so the coefficient of q^∑_i=1^+ d_i α_i in (<ref>) is (-1)^+2 times the coefficient of  q^∑_i=1^+α_i (d_i ++-1)  + ∑_i=1^α_i N in (<ref>). Since 
    ∑_ i=1^+α_i (d_i ++-1)  + ∑_i=1^α_i N =  ∑_i=1^(+ -1+N +d_i ) α_i - ∑_i=+1^+ (- d_i - -   +1)α_i
 the two coefficients agree up to a factor of (-1)^∑_i=1^+ d_i + +2 +N.

However, by <ref>, that coefficient vanishes unless  

    +2 = ∑_i=1^(+ -1+N+ d_i )   - ∑_i=+1^+ (-d_i - -   +1)= (+) (+-1) + N   + ∑_i=1^ d_i - ∑_i=+1^+ d_i
 in which case ∑_i=1^+ d_i + +2 +N is even, so either the two coefficients are equal or they are negatives of each other but zero and hence equal anyways.  

 For S ⊆{1,…, +} of size , we have an equality of holomorphic functions on  
    {α_1,…, α_+∈ℂ|Re(α_i)< 1/4 for  i ∈ S, Re(α_i)> - 1/4 for  i ∉ S }
 given by 
    ∏_1 ≤ i_1 < i_2 ≤+ (q^-α_i_1 - q^ - α_i_2 )  ∏_i∈ S  q^ - α_iN ∑_ f_1,…, f_+∈𝔽_q[u]^+ 
    ∏_i∈ S  f_i = ∏_i∉ S f_i∏_i∈ Sf_i ^ -1/2+α_i  ∏_i∉ Sf_i ^ -1/2-α_i

    =  ∑_ e_1,… e_+∈ℤ
     N ≥ e_1≥… e_
     e_+1≥…  e_+≥ 0   ∑_σ∈ S_+
    σ^-1 (S) = {1,…, }(σ) ∏_i=1^+ q^ - (e_i ++- i)  α_σ(i) ∫_ψ_𝐞μ_ep.
 

  We let τ be any fixed permutation in S_+ with τ^-1 (S) ={1,…, } and perform the change of variables replacing α_i with α_τ(i) in the statement of <ref>, obtaining


    ∏_1 ≤ i_1 < i_2 ≤+ (q^-α_τ(i_1) - q^ - α_τ(i_2) )  ∏_i=1^  q^ -α_τ(i) N ∑_ f_1,…, f_+∈𝔽_q[u]^+ 
    ∏_i=1^ f_i = ∏_i=+1^+ f_i∏_i=1^f_i ^ -1/2+ α_τ(i)∏_i=+1^+f_i ^ -1/2-α_τ(i)

    =  ∑_ e_1,… e_+∈ℤ
     N ≥ e_1≥… e_
     e_+1≥…  e_+≥ 0   ∑_σ∈ S_× S_(σ) ∏_i=1^+ q^ - (e_i ++- i)  α_τ( σ(i) )∫_ψ_𝐞μ_ep.

We have

    ∏_i=1^  q^ -α_τ(i) N   =   ∏_i∈ S  q^ - α_iN
 and 

    ∑_ f_1,…, f_+∈𝔽_q[u]^+ 
    ∏_i=1^ f_i = ∏_i=+1^+ f_i∏_i=1^f_i ^ -1/2-+ α_τ(i)∏_i=+1^+f_i ^ -1/2-α_τ(i) = ∑_ f_1,…, f_+∈𝔽_q[u]^+ 
    ∏_i∈ S  f_i = ∏_i∉ S f_i∏_i∈ Sf_i ^ -1/2+α_i  ∏_i∉ Sf_i ^ -1/2-α_i
 
using the change of variables f_i ↦ f_τ(i), and we have

    ∏_1 ≤ i_1 < i_2 ≤+ (q^-α_τ(i_1) - q^ - α_τ(i_2) )  = (τ)  ∏_1 ≤ i_1 < i_2 ≤+ (q^-α_i_1 - q^ - α_i_2 )
 
so (<ref>) is equal to (τ) times (<ref>). Similarly, the change of variables σ→τ^-1σ gives

    ∑_σ∈ S_× S_(σ) ∏_i=1^+ q^ - (e_i ++- i)  α_τ( σ(i) ) = ∑_τ^-1σ∈ S_× S_(τ^-1σ) ∏_i=1^+ q^ - (d_i ++- i)  α_σ(i)
 
    = (τ) ∑_σ∈ S_+
    σ^-1 (S) = {1,…, }(σ) ∏_i=1^+ q^ - (e_i ++- i)  α_σ(i)

so (<ref>) is equal to (τ) times (<ref>). Thus (<ref>) and (<ref>) are equal to each other. 



 
    ∑_ e_1,… e_+∈ℤ
     N ≥ e_1≥… e_
     e_+1≥…  e_+≥ 0   κ_𝐞∫_ψ_𝐞μ_ep =  MT_N^, .
  


 Since the sum ∑_ e_1,… e_+∈ℤ
 N ≥ e_1≥… e_
 e_+1≥…  e_+≥ 0   κ_𝐞∫_ψ_𝐞μ_ep is uniformly convergent as a function of α_1,…, α_+ on the imaginary axis, both sides are continuous functions of α_1,…, α_+. So it suffices to prove this identity after restricting to  a dense subset, and therefore suffices to prove it after multiplying by the Vandermonde ∏_1 ≤ i_1 < i_2 ≤+ (q^-α_i_1 - q^ - α_i_2 ).  When we do this, the right-hand side becomes, by definition of MT,

    ∏_1 ≤ i_1 < i_2 ≤+ (q^-α_i_1 - q^ - α_i_2)∏_i=+ 1^+ q^α_i N∑_  S ⊆{1,…,+}
    S=∏_i ∈ S q^ -α_i N∑_ f_1,…, f_+∈𝔽_q[u]^+ 
    ∏_i ∈ S f_i = ∏_i∉ S f_i∏_i∈ Sf_i ^ -1/2+α_i  ∏_i ∉ Sf_i ^ -1/2-α_i

which is the sum over S of (<ref>) multiplied by ∏_i=+1^+ q^α_i N. The left-hand side becomes, by definition of κ,

    ∑_ e_1,… e_+∈ℤ
     N ≥ e_1≥… e_
     e_+1≥…  e_+≥ 0     (-1)^N ∑_σ∈ S_+(σ) ∏_i=1^+ q^ - (e_i ++- i)  α_σ(i) ∏_i=+1^+ q^   Nα_i  ∫_ψ_𝐞μ_ep
 

    = ∏_i=+1^+ q^   Nα_i  ∑_  S ⊆{1,…,+}
    S=∑_ e_1,… e_+∈ℤ
     N ≥ e_1≥… e_
     e_+1≥…  e_+≥ 0     (-1)^N ∑_σ∈ S_+
    σ^-1 (S) = {1,…,}(σ) ∏_i=1^+ q^ - (e_i ++- i)  α_σ(i) ∫_ψ_𝐞μ_ep

which is the sum over S of (<ref>) multiplied by ∏_i=+1^+ q^α_i N, since each permutation σ sends {1,…,} to exactly one set S of cardinality . (The rearrangement of the sum is justified by absolute convergence.) The claim then follows from <ref>.


 This follows from combining <ref> and (<ref>) with <ref> and <ref>, noting the error term O ( N^ O(1) q^ - k/4 ) of <ref> is easily absorbed into the error term O (N^  (+)^2/2 - βq-2/4) of <ref> since k = ⌊ N^β⌋ so the exponential term  q^ - k/4 dominates any polynomial in N. 

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