# Ordering of Number Fields and Distribution of Class Groups: Abelian
Extensions

Weitong Wang  Shing-Tung Yau Center of Southeast University, 15th floor Yifu
Architecture Building, 2 Sipailou, Nanjing, Jiangsu Province 210096 China
<EMAIL_ADDRESS>
###### Abstract.

When $p$ divides the ordering of Galois group, the distribution of
$\mathbb{Z}_{p}\otimes\operatorname{Cl}_{K}$ is closely related to the problem
of counting fields with certain specifications. Moreover, different orderings
of number fields affect the answers of such questions in a nontrivial way. So,
in this paper, we set up an invariant of number fields with parameters, and
consider field counting problems with specifications while the parameters
change as a variable. The case of abelian extensions shows that, roughly
speaking, the result of counting abelian fields is a function of the invariant
with parameters.

###### Key words and phrases:

Cohen-Lenstra heuristics, distribution of class groups, field counting

##  1\. Introduction

For an extension $K/k$ of number fields, we denote its Galois closure by
$\hat{K}$. Let’s first present the following conjecture, which is one of the
main statements of [7]. Note that our statement differs by the original one a
little for the purpose of applications in this paper.

######  Definition 1.1.

Let $1\leq G\leq S_{n}$ be a transitive permutation group, and let $k$ be a
number field. Let $\mathcal{S}$ be the set of all number fields $(K/k,\psi)$
such that its Galois closure $(\hat{K}/k,\psi)$ is a $G$-extension (see
Definition 2.2), and that $K=\hat{K}^{H}$ where $H=\operatorname{Stab}_{G}(1)$
is the stabilizer of $1$. Suppose that $\Omega$ is a (nonempty) subset of $G$
that is closed under invertible powering, i.e., if $g^{a}=h$, $h^{b}=g$, then
$g\in\Omega$ if and only if $h\in\Omega$. Define for the set $\Omega$, and for
all $\gamma=1,2,\dots$,

$\mathbf{1}_{(\Omega,\gamma)}(K):=\left\\{\begin{aligned} &1\quad&\text{if
there are exactly }\gamma\text{ rational primes }p\nmid\lvert G\rvert\\\
&&\text{s.t. }I(p)\cap\Omega\neq\emptyset;\\\
&0\quad&\textnormal{otherwise.}\end{aligned}\right.$

where $I(p)$ here means the inertia subgroup of $p$. When $\gamma=0$, define
$\mathbf{1}_{(\Omega,0)}(K)=1$ if there are exactly $\gamma$ rational primes
$p$ such that $I(p)\cap\Omega\neq\emptyset$, and
$\mathbf{1}_{(\Omega,0)}(K)=0$ otherwise.

######  Conjecture 1.2.

Keep $G,k,\mathcal{S}$ as above. Let $\operatorname{id}\notin\Omega$ be a
(nonempty) subset of $G$ that is closed under invertible powering, and let
$\vartheta$ be an invariant of number fields.

  1. (1)

For all $\gamma=0,1,2,\dots$, there exists some $\gamma^{\prime}$, such that

$\sum_{\begin{subarray}{c}K\in\mathcal{S}\\\
\vartheta(K)<x\end{subarray}}\mathbf{1}_{(\Omega,\gamma)}(K)=o\left(\sum_{\begin{subarray}{c}K\in\mathcal{S}\\\
\vartheta(K)<x\end{subarray}}\mathbf{1}_{(\Omega,\gamma^{\prime})}(K)\right),$

In this case we say that the conjecture 1 holds for
$(({\mathcal{S}},\vartheta),\Omega)$.

  2. (2)

For all $\gamma=0,1,2,\dots$,

$\sum_{\begin{subarray}{c}K\in\mathcal{S}\\\
\vartheta(K)<x\end{subarray}}\mathbf{1}_{(\Omega,\gamma)}(K)=o\left(\sum_{\begin{subarray}{c}K\in\mathcal{S}\\\
\vartheta(K)<x\end{subarray}}1\right)$

In this case we say that the conjecture 2 holds for the pair
$(({\mathcal{S}},\vartheta),\Omega)$.

It is worth mentioning that whenever the conjecture holds, we can prove some
results on the distribution of class groups. To be precise, we have the
following.

######  Definition 1.3.

Let $1\leq G\leq S_{n}$ be a finite transitive permutation group. Let
$\sigma\in G$ be any permutation. Define
$e(\sigma):=\gcd(\lvert\langle\sigma\rangle\cdot
1\rvert,\dots,\lvert\langle\sigma\rangle\cdot n\rvert)$, i.e., the greatest
common divisor of the size of orbits. We call $q$ a _non-random prime_ for $G$
if $q\mid e(\sigma)$ for some $\sigma\in G$. On the other hand, for a
permutation $\sigma\in G$, if $q^{l}\|e(\sigma)$, then we call $\sigma$ an
element inertia of type $q^{l}$. Define $\Omega(G,q^{l})$ to be the subset of
$G$ consisting of all elements inertia of type $q^{l}$. We denote
$\bigcup_{l=1}^{\infty}\Omega(G,q^{l})$ by $\Omega(G,q^{\infty})$.

######  Theorem 1.4.

[7] Let $1\leq G\leq S_{n}$ be a transitive permutation group, and let $k$ be
a number field. Let $\mathcal{S}$ be the set of all number fields $(K/k,\psi)$
such that its Galois closure $(\hat{K}/k,\psi)$ is a $G$-extension, and that
$K=\hat{K}^{G_{1}}$. Let $H\subseteq G$ be a subgroup such that
$\hat{K}^{H}\subseteq K$ for $K\in\mathcal{S}$. If $q$ is a non-random prime
for $G$ such that $q\mid[K:\hat{K}^{H}]$, and Conjecture 1.2(2) holds for
$((\mathcal{S},\vartheta),\Omega)$, where $\Omega:=\Omega(G,q^{\infty})$, then

$\mathbb{P}(\operatorname{rk}_{q}\operatorname{Cl}(K/\hat{K}^{H})\leq
r)=0\quad\text{and}\quad\mathbb{E}(\lvert\operatorname{Hom}(\operatorname{Cl}(K/\hat{K}^{H}),C_{q})\rvert)=+\infty,$

where $K$ runs over fields in $\mathcal{S}$ for the product of ramified primes
in $K/\mathbb{Q}$, and $\operatorname{Cl}(K/\hat{K}^{H})$ denotes the relative
class group.

The following example explains that the conjecture may be false if we count
number fields by discriminant.

######  Example 1.5 (Ordering by discriminant).

Let’s consider non-Galois cubic extensions the set
$\mathcal{S}:=\mathcal{S}(S_{3},\\{1,(23)\\})$ of $K/\mathbb{Q}$. In this case
we want to show that the analogous statement of Conjecture 1.2(1) is _false_
for $(\mathcal{S};\\{(123),(132)\\})$ when the fields ordered by
_discriminant_ , despite the fact that $3$ is non-random for $S_{3}$.

According to the work of Bhargava, Shankar, and Tsimerman [1, Theorem 8], we
know that counting nowhere totally ramified degree $3$ cubic fields will give
the main term $cx$ where $c$ is a nonzero constant, which is the same main
term as counting all cubic fields by discriminant. This already contradicts
the analogous statement of Conjecture 1.2(2) when ordering fields by
discriminant, the weaker one. So we can conclude that when cubic fields
ordered by discriminant with non-random prime $3$ is a counter-example for the
analogous result of the conjecture. On the other hand, [7, Proposition 6.5]
shows that, under some hypothesis, Conjecture 1.2(2) holds for
$(\mathcal{S};\\{(123),(132)\\})$. This shows that the conjecture, and
possibly the statistical behaviors of non-random primes that show
qualitatively difference from good primes, are dependent on the choice of the
ordering of the number fields.

So, we will discuss the relation between ordering of number fields and field
counting in this paper. We use an example to show the main result of this
paper. For general case, see Theorem 5.6 and 5.13.

######  Proposition 1.6.

Let $G=C_{2}\times C_{2}$ with generators $g_{1},g_{2}$, and let
$\mathcal{S}:=\mathcal{S}(G)$ be the set of $G$-fields. For each
$K\in\mathcal{S}$, let $e_{1}$ be the product of ramified primes such that
$g_{1}\in I(p)$, where $I(p)$ is the inertia subgroup of $p$. For $t>0$,
define $\vartheta_{t}(K):=e_{1}e_{2}^{t}$ where $e_{2}=P(K)/e_{1}$ where
$P(K)$ is the product of ramified primes in $K/\mathbb{Q}$. Let
$\Omega=\\{g_{2},g_{1}g_{2}\\}$. When $\gamma\geq 1$, we have

$\sum_{\begin{subarray}{c}K\in\mathcal{S}\\\
\vartheta_{t}(K)<x\end{subarray}}\mathbf{1}_{(\Omega,\gamma)}(K)\sim\left\\{\begin{aligned}
&c_{1}(t)x\quad&\text{ if }t>1\\\ &c_{2}x(\log\log x)^{\gamma}\quad&\text{ if
}t=1\\\ &c_{3}(t)\frac{x^{t^{-1}}}{\log x}(\log\log x)^{\gamma}\quad&\text{ if
}t<1\end{aligned}\right.$

where $c_{2}>0$ is a constant, and $c_{1}$, resp. $c_{3}$, is a continuous
function defined in $(1,+\infty)$, resp. in $(0,1)$. Moreover, the Conjecture
1.2(2) holds for $(\mathcal{S},\vartheta_{t})$ if and only if $t\leq 1$.

## Acknowledgement

The author would like to thank Yuan Liu for many useful conversations. This
work was done with the support of Jiangsu Funding Program for Excellent
Postdoctoral Talent.

##  2\. Basic notations

In this section we introduce some of the notations that will be used in the
paper. We use some standard notations coming from analytic number theory. For
example, write a complex number as $s=\sigma+it$. The notation $q^{l}\|n$
means that $q^{l}\mid n$ but $q^{l+1}\nmid n$. Denote the Euler function by
$\phi(n)$. Let $\omega(n)$ counts the number of distinct prime divisors of $n$
and so on.

We also follow the notations of inequalities with unspecified constants from
Iwaniec and Kowalski [4, Introduction, p.7]. Let’s just write down the ones
that are important for us. Let $X$ be some space, and let $f,g$ be two
functions. Then $f(x)\ll g(x)$ for $x\in X$ means that $\lvert f(x)\rvert\leq
Cg(x)$ for some constant $C\geq 0$. Any value of $C$ for which this holds is
called an implied constant. We use $f(x)\asymp g(x)$ for $x\in X$ if $f(x)\ll
g(x)$ and $g(x)\ll f(x)$ both hold with possibly different implied constants.
We say that $f=o(g)$ as $x\to x_{0}$ if for any $\epsilon>0$ there exists some
(unspecified) neighbourhood $U_{\epsilon}$ of $x_{0}$ such that $\lvert
f(x)\rvert\leq\epsilon g(x)$ for $x\in U_{\epsilon}$.

For the convenience of stating the results, we make the following notation.

######  Definition 2.1.

Let $\mathcal{P}$ be the set of rational prime numbers, and for each pair
$(m,n)$ of coprime numbers, we define

$\mathcal{P}(n):=\\{p\in\mathcal{P}\mid\gcd(p,n)=1\\}\text{ and
}\mathcal{P}(m,n):=\\{p\in\mathcal{P}\mid p\equiv m\bmod{n}\\}.$

Since there are multiple ways to describe field extensions, we give the
following two definitions to make the term like “the set of all non-Galois
cubic number fields” precise.

######  Definition 2.2.

For a field $k$, by a _$\Gamma$ -extension of $k$_, we mean an isomorphism
class of pairs $(K,\psi)$, where $K$ is a Galois extension of $k$, and
$\psi:\operatorname{Gal}(K/k)\cong\Gamma$ is an isomorphism. An isomorphism of
pairs $(\tau,m_{\tau}):(K,\psi)\to(K^{\prime},\psi^{\prime})$ is an
isomorphism $\tau:K\to K^{\prime}$ such that the map
$m_{\tau}:\operatorname{Gal}(K/k)\to\operatorname{Gal}(K^{\prime}/k)$ sending
$\sigma$ to $\tau\circ\sigma\circ\tau^{-1}$ satisfies $\psi^{\prime}\circ
m_{\tau}=\psi$. We sometimes leave the $\psi$ implicit, but this is always
what we mean by a $\Gamma$-extension. We also call $\Gamma$-extensions of
$\mathbb{Q}$ _$\Gamma$ -fields._

######  Definition 2.3.

Let $G\subseteq S_{n}$ whose action on $\\{1,2,\dots,n\\}$ is transitive. Let
$k$ be a number field. Let ${\mathcal{S}}(G;k)$ be the set of pairs $(K,\psi)$
such that the Galois closure $(\hat{K},\psi)$ is a $G$-field and that
$K=\hat{K}^{H}$, where $H=\operatorname{Stab}_{G}(1)$ is the stabilizer of
$1$. In other words $\psi$ defines the Galois action of $G$ on $K$ over $k$
and $[K:k]=n$. If the base field $k=\mathbb{Q}$, then we just omit it and
write ${\mathcal{S}}(G):={\mathcal{S}}(G,\mathbb{Q})$.

Then, we give the notation of counting number fields.

######  Definition 2.4.

Let $\mathcal{S}=\mathcal{S}(G;k)$ where $G$ is a transitive permutation
group, and let $\vartheta:\mathcal{S}\to\mathbb{R}^{+}$ be an invariant of
number fields (e.g. discriminant or product of ramified primes). Define

$N(\mathcal{S},\vartheta;x):=\sum_{K\in\mathcal{S},\vartheta(K)<x}1.$

If $f$ is a function defined over $\mathcal{S}$, then we define

$N(\mathcal{S},\vartheta;f;x):=\sum_{K\in\mathcal{S},\vartheta(K)<x}f(K).$

In particular, if $f=\mathbf{1}_{(\Omega,r)}$ (see Definition 1.1), then just
write

$N(\mathcal{S},\vartheta;(\Omega,r);x):=N(\mathcal{S},\vartheta;\mathbf{1}_{(\Omega,r)};x).$

##  3\. Invariant of number fields

Now let $G$ be a transitive permutation group in $S_{n}$. For any number field
$K\in\mathcal{S}(G)$, let $\hat{K}$ be its Galois closure, i.e.,
$\hat{K}/\mathbb{Q}$ is the smallest field extension containing $K$. Note that
$G$ is actually the Galois group of $\hat{K}/\mathbb{Q}$ according to its
action on the embeddings $K\to\bar{\mathbb{Q}}$ in the definition of
$\mathcal{S}(G)$. In general, the extension $K/\mathbb{Q}$ is not normal, so
for any rational prime $p$, let $G(p)$ be its decomposition group in $G$ up to
conjugation. In particular, $G(p)$ will determine the factorization of
$p\mathscr{O}_{K}$. Since decomposition group admits a normal subgroup, the
inertia subgroup, we define $S$ to be the set of all ordered pairs of
subgroups $(H_{1},H_{2})$ such that $H_{1}\triangleleft H_{2}$. Then, define a
map $V:S\to\mathbb{R}_{\geq 0}$, such that $V(\\{\operatorname{id}\\},H)=0$
for any $H\subseteq G$. For each rational prime $p$, let
$\vartheta(K_{p}):=p^{V(I(p),G(p))}$, where $K_{p}=\mathbb{Z}_{p}\otimes K$
and $I(p)$ is the inertia subgroup (up to conjugation). By abuse of notation,
let

$\vartheta(K):=\prod_{p\in\mathcal{P}}\vartheta(K_{p}),$

where $\mathcal{P}$ is the set of all rational primes. This construction of an
invariant $\vartheta$ for fields $K\in\mathcal{S}(G)$ is actually more than we
need. Recall that we have defined a function $\mathbf{1}_{(\Omega,\gamma)}$ in
Definition 1.1, where $\Omega$ is some subset of $G$ closed under invertible
powering and conjugation. It is enough to consider the invariants $\vartheta$
based on $V$ such that $V(H_{1},G_{1})=V(H_{2},G_{2})$ if $H_{1}=H_{2}$, i.e.,
$V$ is actually a map $\\{H\mid H\subseteq G\\}\to\mathbb{R}_{\geq 0}$.

On the other hand, this construction is abstract and not convenient in the
applications. So, we first study the property of subsets $\Omega\subseteq G$
that are closed under invertible powering and conjugation. Recall that
$\Omega$ is closed under invertible powering if for any $g,h\in G$ such that
$g^{a}=h$ and $h^{b}=g$ for some $a,b\in\mathbb{Z}$, we have $g\in\Omega$ if
and only if $h\in\Omega$.

######  Definition 3.1.

Let $G$ be a finite transitive group. Given two elements $g_{1},g_{2}\in G$,
if there exists some $a,b\in\mathbb{Z}$ and some $h_{1},h_{2}\in G$ such that

$g_{1}^{a}=h_{2}g_{2}h_{2}^{-1}\quad\text{and}\quad
g_{2}^{b}=h_{1}g_{1}h_{1}^{-1},$

then we say that $g_{1}$ and $g_{2}$ are equivalent under closed powering and
conjugation, denoted by $g_{1}\sim g_{2}$. If there is no danger of confusion,
we simply say that $g_{1}$ and $g_{2}$ are equivalent.

One can check that this really defines an equivalence relation on $G$. Let’s
consider abelian case as a first example. Let $G$ be an abelian group. Note
that by Class Field Theory, for each
$\rho\in\operatorname{Hom}(\prod_{p}\mathbb{Z}_{p}^{*},G)$, there exists
exactly one Artin reciprocity map $\tilde{\rho}:\mathbf{C}_{\mathbb{Q}}\to G$,
where $\mathbf{C}_{\mathbb{Q}}$ is the idèle class group, such that locally we
have $\tilde{\rho}|_{p}=\rho|_{p}$ for all finite prime $p$ of $\mathbb{Q}$.
See also [3, p.291-339], or the appendix for more details. Now let $\bar{G}$
be the set $G/\sim$ where $g\sim h$ if they are equivalent under invertible
powering. Write $\bar{G}=\\{\\{0\\},\Omega_{1},\Omega_{2},\dots,\Omega_{l}\\}$
where $\Omega_{i}$ is an equivalence class written as a subset of $G$. We can
see that the subgroup $H_{i}:=\langle\Omega_{i}\rangle$ generated by
$\Omega_{i}$ is actually cyclic, and its generator is just any element
$g\in\Omega_{i}$. So, $\bar{G}$ is partially ordered as follows. We say that
$\Omega_{i}\leq\Omega_{j}$ if $H_{i}\subseteq H_{j}$. For each
$K\in\mathcal{S}(G)$, we can define an invariant with parameters
$\underline{x}=(x_{1},\dots,x_{l})\in\mathbb{R}_{+}^{l}$.

######  Definition 3.2.

Let $\rho:\mathbf{C}_{\mathbb{Q}}\to G$ be the Artin reciprocity map
corresponding to $K$. For each rational prime $p$, let $I(p)$ be the image of
$\rho_{p}:\mathbb{Z}_{p}^{*}\to G$. Let
$\overline{I(p)}=\\{\Omega_{i}\subseteq I(p)\\}$ be the set of equivalence
classes in $I(p)$. If $\Omega_{p_{1}},\dots,\Omega_{p_{m}}$ are maximal
elements in $\overline{I(p)}$, then define

$x(H):=x_{p_{1}}+\cdots+x_{p_{m}}.$

And let

$\vartheta_{\underline{x}}(\rho_{p}):=p^{x(I(p))}.$

Finally we define

$\vartheta_{\underline{x}}(K):=\prod_{p\in\mathcal{P}}\vartheta_{\underline{x}}(\rho_{p}).$

Clearly, for tamely ramified primes, the inertia subgroup $I(p)$ is cyclic,
and $\vartheta_{\underline{x}}(\rho_{p})$ just requires finding out the
corresponding index of $I(p)$. Then let’s show the computation of
$\mathbf{1}_{(\Omega,\gamma)}$ in the view of Class Field Theory.

######  Definition 3.3.

Let $\operatorname{id}\notin\Omega$ be a subset of $G$ closed under invertible
powering, and let $\rho\in\operatorname{Hom}(\prod_{p}\mathbb{Z}^{*}_{p},H)$
be a homomorphism. Let
$\rho_{p}\in\operatorname{Hom}(\prod_{p}\mathbb{Z}_{p}^{*},G)$ be a local
morphism where $p$ is a rational prime. Define

$\mathbf{1}^{(p)}_{\Omega}(\rho_{p})=\left\\{\begin{aligned} &1\quad\text{if
}\rho_{p}(\mathbb{Z}_{p}^{*})\cap\Omega\neq\emptyset;\\\
&0\quad\text{otherwise.}\end{aligned}\right.$

When $\gamma>0$, define $\mathbf{1}_{(\Omega,\gamma)}(\rho)=1$ if there are
exactly $\gamma$ rational primes $p\nmid\lvert G\rvert$ such that
$\mathbf{1}^{(p)}_{\Omega}(\rho_{p})=1$, and $0$ otherwise. When $\gamma=0$,
define $\mathbf{1}_{(\Omega,\gamma)}(\rho)=1$ if there are no rational prime
$p$ such that $\mathbf{1}^{(p)}_{\Omega}(\rho_{p})=1$ and $0$ otherwise. If
$\tilde{\rho}\in\operatorname{Hom}(\mathbf{C}_{\mathbb{Q}},G)$ is surjective,
corresponding to the class field $K$, then just define
$\mathbf{1}_{(\Omega,\gamma)}(\tilde{\rho}):=\mathbf{1}_{(\Omega,\gamma)}(K)$.

One can check that if $\tilde{\rho}_{p}=\rho_{p}$ for all $p$ of $\mathbb{Q}$,
then
$\mathbf{1}_{(\Omega,\gamma)}(\tilde{\rho})=\mathbf{1}_{(\Omega,\gamma)}(\rho)$.
This definition makes the functions $\mathbf{1}_{(\Omega,\gamma)}$ and the
invariant function $\vartheta_{\underline{x}}$ more compatible. To be precise,
we have the following statement.

######  Lemma 3.4.

The partially ordered set $\overline{\rho_{p}(\mathbb{Z}_{p}^{*})}$ is just

$\\{\Omega_{i}\in\bar{G}\mid\mathbf{1}_{\Omega_{i}}^{(p)}(\rho_{p})=1\\}.$

For tamely ramified primes, there is a unique maximal element in
$(\overline{\rho_{p}(\mathbb{Z}_{p}^{*})},\leq)$, because the image of
$\mathbb{Z}_{p}^{*}$ under $\rho_{p}$ is cyclic. From the definition, let
$\rho\in\operatorname{Hom}(\prod_{p}\mathbb{Z}_{p}^{*},G)$ such that
$\mathbf{1}_{\Omega}^{(p)}(\rho_{p})=1$ only if $p\nmid\lvert G\rvert$, then

$\vartheta_{\underline{x}}(\rho)=e_{1}^{x_{1}}\cdots e_{l}^{x_{l}},$

where $e_{i}$ is the product of ramified primes $p$ such that the image of
$\rho_{p}$ is exactly the subgroup $H_{i}$, for all $i=1,\dots,l$. For
example, the product of ramified primes comes from the invariant
$\vartheta_{(1,1,\dots,1)}$ on the set $\mathcal{S}$ of $G$-fields, up to
wildly ramified primes. On the other hand, the discriminant of
$K\in\mathcal{S}$ requires the formula of computing
$\operatorname{Disc}(\rho_{p})$ for each $p$. But it is also true that for
tamely ramified primes, $\operatorname{Disc}(\rho_{p})$ is determined by
$I(p)$, so it is enough to consider parameters associated to $\bar{G}$ as
we’ve mentioned.

######  Example 3.5.

To give a concrete example, let $G\cong C_{6}$ with generator $g$. Then

$S=\\{\\{0\\},\Omega_{0}:=\\{g,g^{5}\\},\Omega_{1}:=\\{g^{2},g^{4}\\},\Omega_{2}:=\\{g^{3}\\}\\}.$

The invariant $\vartheta_{(1,1,1)}$ is exactly the product of ramified primes.
For a tamely ramified prime $p$, we have the following computation: if
$\rho_{p}(\mathbb{Z}_{p}^{*})=\langle\Omega_{0}\rangle$, then
$\operatorname{Disc}(\rho_{p})=p^{5}$; if
$\rho_{p}(\mathbb{Z}_{p}^{*})=\langle\Omega_{1}\rangle$, then
$\operatorname{Disc}(\rho_{p})=p^{4}$; if
$\rho_{p}(\mathbb{Z}_{p}^{*})=\langle\Omega_{2}\rangle$, then
$\operatorname{Disc}(\rho_{p})=p^{3}$. So, $\vartheta_{(5,4,3)}$ is the
discriminant if we ignore wildly ramified primes.

Assume without loss of generality that $H_{i}\cap\Omega\neq\emptyset$ if and
only if $i=1,\dots,m$. If $K/\mathbb{Q}$ is tamely ramified, then

$\mathbf{1}_{(\Omega,\gamma)}(K)=1$

if and only if $\omega(e_{1}\cdots e_{m})=\gamma$, where $\omega$ is the
function counting distinct prime factors of an integer.

Before we get into any concrete examples of counting fields, let’s prove a
lemma here. Provided that $\vartheta$ is an invariant for $\mathcal{S}(G)$,
where $G$ is a transitive permutation group as above, then clearly
$\vartheta^{a}$ where $a>0$ is any real number will also be an invariant.

######  Lemma 3.6.

Let $G$ be a transitive permutation group, let $\Omega$ be a subset of $G$,
and let $\vartheta$ be an invariant of $\mathcal{S}:=\mathcal{S}(G)$. If
$f,g:\mathcal{S}\to\mathbb{R}$ are two functions with some constant
$C\in\mathbb{R}$ or $C=\pm\infty$, such that

(3.1)
$\lim_{X\to\infty}\frac{N(\mathcal{S},\vartheta;f;X)}{N(\mathcal{S},\vartheta;g;X)}=C,$

then for any $a>0$, we also have

$\lim_{X\to\infty}\frac{N(\mathcal{S},\vartheta^{a};f;X)}{N(\mathcal{S},\vartheta^{a};g;X)}=C.$

###### Proof.

We prove the case when $C$ is a constant, and the other cases are left to the
reader, for they are similar to each other. Let $\epsilon>0$ be any positive
number. Then by our condition (3.1), which says that the limit exists, there
exists some $N>0$ such that for all $X>N$ we have

$\left\lvert\frac{N(\mathcal{S}(G),\vartheta;f;X)}{N(\mathcal{S}(G),\vartheta;g;X)}-C\right\rvert<\epsilon.$

Note that for any $X>0$, we have $\vartheta<X$ if and only if
$\vartheta^{a}<X^{a}$. This implies that for all $X>N^{a}$, we also have

$\left\lvert\frac{N(\mathcal{S}(G),\vartheta^{a};f;X)}{N(\mathcal{S}(G),\vartheta^{a};g;X)}-C\right\rvert<\epsilon.$

And we are done for the proof when $C$ is a constant. ∎

In the sense of the Conjecture 1.2, we can see that if the conjecture holds
for $((\mathcal{S},\vartheta),\Omega)$, then it will also hold for
$((\mathcal{S},\vartheta^{a}),\Omega)$. We conclude this section by the
following definition.

######  Definition 3.7.

Let $G$ be a transitive permutation group with a subset $\Omega$ closed under
invertible powering and conjugation. Let $\mathcal{S}:=\mathcal{S}(G)$, and
$\vartheta$ be an invariant with parameters
$\underline{x}=(x_{1},\dots,x_{l})\in\mathbb{R}_{+}^{l}$, define for each
$(\gamma_{1},\gamma_{2})\in\mathbb{N}^{2}$ the function
$R^{\gamma_{1}}_{\gamma_{2}}:\mathbb{R}_{+}^{l}\to\mathbb{R}_{\geq 0}$, given
by the expression

$R^{\gamma_{1}}_{\gamma_{2}}(\underline{x}):=\lim_{X\to\infty}\frac{N(\mathcal{S},\vartheta(\underline{x});(\Omega,\gamma_{1});X)}{N(\mathcal{S},\vartheta(\underline{x});(\Omega,\gamma_{2});X)}.$

Note that $R$ may not be well-defined everywhere on $\mathbb{R}_{+}^{l}$. By
Lemma 3.6, we see that if $R^{\gamma_{1}}_{\gamma_{2}}(x_{1},\dots,x_{l})=c$
for some real number $c\geq 0$, then so is
$R^{\gamma_{1}}_{\gamma_{2}}(x_{1}^{a},\dots,x_{l}^{a})$ for all $a>0$.

##  4\. Tauberian Theorem and Dirichlet series

In this section, we first make a brief introduction to Delange’s Tauberian
Theorem, and explain its application via a result that is used repeatedly in
the rest of this paper. See Delange [2] for more details. The work of Delange,
generalizing Ikehara’s Theorem, could be called “effective” or transcendental
Tauberian theorem. Because unlike Hardy-Littlewood Theorem (see [5, Theorem
5.11] for example) which relates the behaviour of the real function
$\sum_{n}a_{n}n^{\sigma}$ when $\sigma>1$ on the real line to that of
$\sum_{n\leq x}a_{n}/n$, it instead cares about the asymptotic information
about $\sum_{n<x}a_{n}$ using the value of the function $\sum_{n}a_{n}n^{s}$
when $s$ is complex, which looks like a “jump” from the half-plane $\sigma>1$
to the point $s=0$.

We’ll mainly use the following result from Delange’s paper. The condition and
notation is a little different from the original one so that it is more
convenient for applications here.

######  Theorem 4.1.

[2, Théorème III,IV.] Let $A(x)$ be a real function defined on $x\geq 0$, non-
negative and non-decreasing. Assume that the integral

$s\int_{1}^{+\infty}x^{-s-1}A(x)\operatorname{d}x$

is convergent in the half-plane $\sigma>\sigma_{0}$, where $\sigma_{0}$ is a
positive real number, and we denote its value by $f(s)$. Let $a$ be a non-
negative real number. Suppose that the function $f(s)$ is holomorphic at all
points on the line $\Re(s)=\sigma_{0}$ other than $\sigma_{0}$ and in the
neighbourhood of $\sigma_{0}$ we have

(4.1) $f(s)=(s-\sigma_{0})^{-\beta}g(s)+h(s)$

with $g$ and $h$ holomorphic at $\sigma_{0}$ and $g(\sigma_{0})\neq 0$. Then
we have for $x\to+\infty$

$A(x)\sim\frac{g(\sigma_{0})}{\Gamma(\beta)}x^{\sigma_{0}}(\log x)^{\beta-1}.$

If the function $f(s)$ satisfies the same condition, except that in the
neighbourhood of $\sigma_{0}$, we have

(4.2)
$f(s)=(s-\sigma_{0})^{\beta}\sum_{j=0}^{\gamma}g_{j}(s)\left(\log\frac{1}{s-\sigma_{0}}\right)^{j}+h(s)\quad(\gamma\geq
1),$

where the functions $g_{j}$ and the function $h$ are holomorphic at
$\sigma_{0}$, and $g_{\gamma}(\sigma_{0})$ is different from $0$. Then we have
for $x\to+\infty$

$A(x)\sim\left\\{\begin{aligned}
&\frac{g_{\gamma}(\sigma_{0})}{\Gamma(\beta)}x^{\sigma_{0}}(\log
x)^{\beta-1}(\log\log x)^{\gamma}&\text{ if }\beta>0\\\ &\gamma
g_{\gamma}(\sigma_{0})\frac{x^{\sigma_{0}}}{\log x}(\log\log
x)^{\gamma-1}&\text{ if }a=0\end{aligned}\right.$

The theorem is not difficult to apply to the Dirichlet series. If
$f(s)=\sum_{n}a_{n}n^{-s}$ is a Dirichlet series, then $f(s)$ can be expressed
in the integral form, i.e.,

$s\int_{0}^{\infty}x^{-s-1}A(x)\operatorname{d}x,$

where $A(x)$ is just the summatory function $\sum_{n<x}a_{n}$. So, as long as
we can prove the analytic continuation (which is of course not easy in
general) together with the pole behaviour at its abscissa of convergence, then
we can apply the theorem. See also Narkiewicz [6, Appendix II Theorem I] for
example. However, we are looking at objects more than the usual Dirichlet
series in this paper.

We need some notations before stating our result.

######  Definition 4.2.

  1. (i)

Let $\imath:\mathbb{N}\to\mathbb{R}_{\geq 0}$ be an injective increasing map
and let $S\subset\mathbb{R}_{\geq 0}$ be the image of $\imath$. If for each
$N>0$, for all but finitely many $n\in\mathbb{N}$ we have $\imath(n)>N$, then
we say that $(\imath,S)$ is a set of index. When there is no danger of
confusion, we just omit $\imath$ and say $S$ is an index.

  2. (ii)

Let $S$ be a set of index, and let $a:S\to\mathbb{R}_{\geq 0}$ be a map. Then
define

$D_{a}(s):=\sum_{d\in S}a_{d}d^{-s},$

to be its generating series, i.e., $a_{d}=a(d)$ for each $d\in S$.

The first example is of course when $S=\mathbb{N}$, and $D_{a}(s)$ is the
usual Dirichlet series with non-negative real coefficients. Of course, when we
want to emphasize that it is a complex function, we can just write
$f(s)=D_{a}(s)$ In this paper, we’ll usually let capital letters denote the
summatory functions. For example, let

$A(x):=\sum_{d<x}a_{d}.$

It is also common to obtain a series $f(s)=D_{a}(s)$ from some other methods,
say Euler products, without knowing much about the map $a:S\to\mathbb{R}_{\geq
0}$. Since we are mainly interested in the coefficients, i.e., the arithmetic
function $a:S\to\mathbb{R}_{\geq 0}$, let’s make the following definition so
that we can compare the coefficients of generating series.

######  Definition 4.3.

Let $S$ be an index set. Let $D_{a}(s)=\sum_{d\in S}a_{d}d^{-s}$ and
$D_{b}(s)=\sum_{d\in S}b_{d}d^{-s}$ be two generating series with non-negative
real coefficients. We say that $D_{a}(s)\leq D_{b}(s)$ if $a_{d}\leq b_{d}$
for all $d\in S$.

Then we establish the basic properties of the generating series. In the rest
of this section, let $(\imath,S)$ be a fixed set of index, and let
$a:S\to\mathbb{R}_{\geq 0}$ be a map with generating series $D_{a}(s)$. The
first result shows that, just like Dirichlet series, the generating series
$D_{a}(s)$ converges in a half-plane.

######  Proposition 4.4.

Suppose that $D_{a}(s)$ converges at a point $s=s_{0}$, and that $H>0$ is a
positive real number. Then the series $D_{a}(s)$ is uniformly convergent in
the sector $\mathcal{T}:=\\{s:\sigma\geq\sigma_{0},\lvert t-t_{0}\rvert\leq
H(\sigma-\sigma_{0})\\}$.

The proof is similar to the case of Dirichlet series. See [5, Theorem 1.1] for
example.

###### Proof.

Let $R(u):=\sum_{d>u}a_{d}d^{-s_{0}}$ be the remainder term of the series
$D_{a}(s_{0})$. First we show that for any $s$,

(4.3) $\displaystyle\sum_{d=\imath(M+1)}^{\imath(N)}a_{d}d^{-s}=$
$\displaystyle
R(\imath(M))(\imath(M))^{s_{0}-s}-R(\imath(N))(\imath(N))^{s_{0}-s}$
$\displaystyle+$
$\displaystyle(s_{0}-s)\int_{\imath(M)}^{\imath(N)}R(u)u^{s_{0}-s-1}\operatorname{d}u.$

Note that $a_{\imath(j)}=(R(\imath(j-1))-R(\imath(j)))\imath(j)^{s_{0}}$.
According to Riemann-Stieltjes integral and integral by parts, we have

$\displaystyle\sum_{d=\imath(M+1)}^{\imath(N)}a_{d}d^{-s}$
$\displaystyle=-\int_{\imath(M)}^{\imath(N)}u^{s_{0}-s}\operatorname{d}R(u)$
$\displaystyle=-u^{s_{0}-s}R(u)\Big{|}_{\imath(M)}^{\imath(N)}+\int_{\imath(M)}^{\imath(N)}R(u)\operatorname{d}u^{s_{0}-s}$
$\displaystyle=$ $\displaystyle
R(\imath(M))(\imath(M))^{s_{0}-s}-R(\imath(N))(\imath(N))^{s_{0}-s}$
$\displaystyle+$
$\displaystyle(s_{0}-s)\int_{\imath(M)}^{\imath(N)}R(u)u^{s_{0}-s-1}\operatorname{d}u.$

If $\lvert R(u)\rvert\leq\varepsilon$ for all $u\geq M$ and if
$\sigma\geq\sigma_{0}$, then from (4.3) we see that

$\left\lvert\sum_{d=\imath(M+1)}^{\imath(N)}a_{d}d^{-s}\right\rvert\leq
2\varepsilon+\varepsilon\lvert
s-s_{0}\rvert\int_{M}^{\infty}u^{\sigma_{0}-\sigma-1}\operatorname{d}u\leq\left(2+\frac{\lvert
s-s_{0}\rvert}{\sigma-\sigma_{0}}\right)\varepsilon.$

For $s$ in the prescribed region we see that

$\lvert s-s_{0}\rvert\leq\sigma-\sigma_{0}+\lvert
t-t_{0}\rvert\leq(H+1)(\sigma-\sigma_{0}),$

so that the sum $\sum_{d=\imath(M+1)}^{\imath(N)}a_{d}d^{-s}$ is uniformly
small, and the result follows from the uniform version of Cauchy principle. ∎

######  Remark.

Proposition 4.4 implies that the generating series $D_{a}(s)$ converges in a
half-plane $\sigma\geq\sigma_{c}$, and we call it _abscissa of convergence_.
Moreover, since each term of the series is a regular function (complex
analytic function or holomorphic function) in the open half-plane $\sigma>0$
and the series itself is locally uniformly convergent in $\sigma>\sigma_{c}$,
we see that series $D_{a}(s)$ is also a regular function in
$\sigma>\sigma_{c}$ by Weierstrass principle.

The following result is an analogous statement of Dirichlet series, saying
that $D_{a}(s)$ could be expressed in the form of an integral using the idea
of Riemann-Stieltjes integral. See [5, Theorem 1.3] for example.

######  Proposition 4.5.

Let $A(x):=\sum_{d<x}a_{d}$ be the summatory function. Denote the abscissa of
convergence of $D_{a}(s)$ by $\sigma_{c}$. If $\sigma_{c}<0$, then $A(x)$ is
bounded and

(4.4) $D_{a}(s)=s\int_{0}^{\infty}x^{-s-1}A(x)\operatorname{d}x,$

for $\sigma>0$. If $\sigma_{c}\geq 0$, then

(4.5) $\limsup_{x\to\infty}\frac{\log\lvert A(x)\rvert}{\log x}=\sigma_{c},$

and (4.4) holds for $\sigma>\sigma_{c}$.

###### Proof.

Let’s do the following computation:

$\displaystyle\sum_{n<N}a_{d}d^{-s}$
$\displaystyle=\int^{N}_{1^{-}}x^{-s}\operatorname{d}A(x)$
$\displaystyle=x^{-s}A(x)\big{|}_{0^{-}}^{N}-\int_{1^{-}}^{N}x^{-s-1}A(x)\operatorname{d}x^{-s}$
$\displaystyle=A(N)N^{-s}+s\int_{1^{-}}^{N}x^{-s-1}A(x)\operatorname{d}t.$

Let $\alpha$ be the left-hand side of (4.5). If $\beta>\alpha$, then this says
that $A(x)\ll x^{\beta}$ as $x\to\infty$ where the implied constant may depend
on $a_{n}$ and $\beta$. For a complex number $s$, if $\sigma>\beta$, then the
integral (4.4) is absolutely convergent, and

$\lim_{N\to\infty}A(N)N^{-s}=0.$

According to Proposition 4.4, if $\sigma_{c}<0$, then
$\lim_{x\to\infty}A(x)=D_{a}(0)$ is a finite number, hence $A(x)$ must be
bounded. So the statement holds when $\sigma_{c}<0$.

If $\sigma_{c}\geq 0$, then by Proposition 4.4, we know that for any
$\beta<\sigma_{c}$, the series $D_{a}(s)$ is divergent. So, $\alpha$ has to be
$\geq\sigma_{c}$, otherwise the above computation would imply that $D_{a}(s)$
is convergent at some point such that $\sigma<\sigma_{c}$. On the other hand,
choose any $\sigma_{0}>\sigma_{c}$. By (4.3) with $s=0$ and $M=0$, we have

$A(\imath(N))=-R(\imath(N))(\imath(N))^{\sigma_{0}}+\sigma_{0}\int_{0}^{\imath(N)}R(u)u^{\sigma_{0}-1}\operatorname{d}u.$

Since $R(u)$ is a bounded function, we know that

$A(x)\ll x^{\sigma_{0}}$

as $x\to\infty$, where the implied constant may depend on $a_{n}$ and on
$\sigma_{0}$. This shows that $\alpha\leq\sigma_{0}$, hence
$\alpha\leq\sigma_{c}$. ∎

These two theorems establish the connection between generating series and
Delange’s Theorem. Roughly speaking, as long as we can show that a generating
series $D_{a}(s)$ admits an analytic continuation, then it satisfies the
condition of Tauberian Theorem, hence we know an estimate for the summatory
function $A(x)$. Just like Dirichlet series, a generating series $D_{a}(s)$
also admits the concept of absolute convergence. To be precise, we say that a
generating series $D_{a}(s)=\sum_{d\in S}a_{d}d^{-s}$ is _absolutely
convergent_ if

$\sum_{d\in S}\lvert a_{d}\rvert d^{-s}$

is convergent. Let $\sigma_{a}$ be the _abscissa of absolute convergence_.

######  Lemma 4.6.

If $\delta$ is the abscissa of convergence of the series

$\sum_{d}d^{-s},$

then $\sigma_{c}\leq\sigma_{a}\leq\sigma_{c}+\delta$.

###### Proof.

The inequality $\sigma_{c}\leq\sigma_{a}$ is clear. Let $\epsilon>0$ be a
positive real number. Since $\sum_{d}a_{d}d^{-\sigma_{c}-\epsilon}$ is
convergent, we see that $a_{d}\ll d^{\sigma_{c}+\epsilon}$ as $d\to\infty$,
and the implied constant may depend on $a_{d}$ and $\epsilon$. So the series
$\sum_{d}\lvert a_{d}\rvert d^{-\sigma_{c}-2\epsilon-\delta}$ must be
convergent by comparing with

$\sum_{d}d^{-\delta-\epsilon}.$

∎

We prove some results that will be used later. In the rest of the section, let
$\underline{x}\in\mathbb{R}_{+}^{l}$ denote an $l$-dimensional vector with
positive coordinates, and let

$S(\underline{x}):=\\{d\in\mathbb{R}_{+}\mid
d=e_{1}^{x_{1}}e_{2}^{x_{1}+x_{2}}\cdots e_{l}^{x_{1}+\cdots+x_{l}}\text{
where }e_{i}\text{ is a positive integer}\\}$

be the index set with parameter $\underline{x}$. Denote by $\mathbb{H}_{a}$
the open half-plane $\sigma>a$ in $\mathbb{C}$. Denote by $\underline{u}^{n}$
a vector of dimension $n$, and omit the superscript $n$ if there is no danger
of confusion.

######  Definition 4.7.

Let $U_{0}\subseteq\mathbb{R}_{+}^{l}\times\mathbb{H}_{0}$ be a neighbourhood
(a subset containing open subset). Let $N>0$ be a positive integer number. We
say that a map $b:\mathcal{P}\to\mathbb{R}_{+}$ is a function modulo $N$ if
$b(p)=b(q)$ whenever $p\equiv q\bmod{N}$ for all rational primes
$p,q\in\mathcal{P}$. A function $a:\mathcal{P}\times U_{0}\to\mathbb{C}$ is
called a coefficient with main term $b:\mathcal{P}\to\mathbb{R}_{+}$ if there
exists $\delta>0$ such that for each rational prime $p\in\mathcal{P}$, we have

$a(p,\underline{x},s)-b(p)\ll p^{-\delta\sigma}$

for $(\underline{x},s)\in U_{0}$, where the implied constant is independent of
$p$.

######  Proposition 4.8.

Let $U=\mathbb{R}_{+}^{l}\times\mathbb{H}_{0}$ be an open subset of
$\mathbb{R}^{l}\times\mathbb{C}$, and let $a:\mathcal{P}\times U\to\mathbb{C}$
be a coefficient with main term $b$, where $0\neq
b:\mathcal{P}\to\mathbb{R}_{+}$ is a non-negative function modulo $N$. Define

$D(\underline{x},s):=\prod_{p\in\mathcal{P}}\big{(}1+\sum_{i=1}^{l}a(p,\underline{x},s)p^{-x_{1}s}\big{)}.$

Then $D(\underline{x},s)$ defines a continuous function in a subset

$U_{0}:=\bigcup_{\underline{x}\in\mathbb{R}_{+}^{l}}\\{\underline{y}\in\mathbb{R}_{+}^{l}\mid
y_{1}\geq x_{1}\\}\times\mathbb{H}_{x_{1}^{-1}},$

of $U$, and for each $\underline{x}\in\mathbb{R}_{+}^{l}$, the series
$D(s):=D(\underline{x},s)$ defines a regular function in the open-half plane
$\mathbb{H}_{x_{1}^{-1}}$. Moreover, there exists some continuous function
$f(\underline{x},s):\bar{U}_{0}\to\mathbb{C}$ such that
$f(\underline{x},s)\neq 0$ for each $\underline{x}\in\mathbb{R}_{+}^{l}$ and
$\Re(s)\geq x_{1}^{-1}$, and such that

$D(\underline{x},s)^{\phi(N)}=f(\underline{x},s)(s-\frac{1}{x_{1}})^{-\beta}$

where $\beta$ is a natural number given by

$\beta=\sum_{\begin{subarray}{c}n\in(\mathbb{Z}/N)^{*}\\\ p\equiv
n\bmod{N}\end{subarray}}b(p).$

###### Proof.

According to the definition of the coefficient $a$, let $C>0$ be a real number
such that for each $p$ and
$(\underline{x},s)\in\mathbb{R}_{+}^{l}\times\mathbb{H}_{0}$ we have

$\lvert b(p)\rvert<C\text{ and }\lvert a(p,\underline{x},s)\rvert<C$

for all $p\in\mathcal{P}$ and
$(\underline{x},s)\in\mathbb{R}_{+}^{l}\times\mathbb{H}_{0}$. By the
inequality

$\lvert
D(\underline{x},s)\rvert\leq\prod_{p\in\mathcal{P}}(1+Cp^{-x_{1}\sigma}),$

we see that $D(\underline{x},s)$ defines a continuous function in $U_{0}$. For
each $\underline{x}\in\mathbb{R}_{+}^{l}$, the function
$D(s):=D(\underline{x},s)$ is a regular function in $\mathbb{H}_{x_{1}^{-1}}$.
To see this result, we can write

$D(\underline{x},s)=\sum_{d\in
S(\underline{x})}a_{d}(\underline{x},s)d(\underline{x})^{-s}$

for each $\underline{x}$. Since $D(s)$ and

$\lvert\sum_{d\in
S(\underline{x})}a_{d}(\underline{x},s)d(\underline{x})^{-s}\rvert\leq\sum_{d}\lvert
a_{d}(\underline{x})\rvert\lvert d(\underline{x})^{-s}\rvert$

are both absolutely convergent in $\mathbb{H}_{x_{1}^{-1}}$, hence equal to
each other. So $D(\underline{x},s)$ must be a regular function in
$\mathbb{H}_{x_{1}^{-1}}$. Let

$\tilde{D}(\underline{s}):=\prod_{p\in\mathcal{P}}\big{(}1+b(p)p^{-x_{1}s}\big{)}.$

Similar argument implies that $\tilde{D}(\underline{x},s)$ is also continuous
in $U_{0}$, and for each $\underline{x}\in\mathbb{R}_{+}^{l}$, the function
$\tilde{D}(\underline{x},s)$ is a regular function in
$\mathbb{H}_{x_{1}^{-1}}$. According to our assumption on the coefficient
function $a$, we may assume without loss of generality that
$\tilde{D}(\underline{s}),D(\underline{s})$ are both non-zero in $U$.
Otherwise, we only take into account large enough $p$. For example, replace
$\tilde{D}(\underline{s})$ by

$\prod_{p>C}\bigl{(}1+b(p)p^{-x_{1}s}\bigr{)}.$

Then take logarithmic and we have

$\displaystyle\lvert\log
D(\underline{x},s)-\log\tilde{D}(\underline{x},s)\rvert$ $\displaystyle=$
$\displaystyle\Big{\lvert}\sum_{p\in\mathcal{P}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}a(p,\underline{x},s)^{n}p^{-nx_{1}s}-\sum_{p\in\mathcal{P}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}b(p)^{n}p^{-nx_{1}s}\Big{\rvert}$
$\displaystyle=$
$\displaystyle\Big{\lvert}\sum_{p\in\mathcal{P}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}\Bigl{(}a(p,\underline{x},s)^{n}p^{-nx_{1}s}-b(p)^{n}p^{-nx_{1}s}\Bigr{)}\Big{\rvert}$
$\displaystyle\ll$
$\displaystyle\sum_{p\in\mathcal{P}}p^{-(x_{1}+\delta)\sigma}+\sum_{p\in\mathcal{P}}\sum_{n=2}^{\infty}\frac{C^{n}}{n}p^{-nx_{1}\sigma}$

This implies that the series of the difference $\log
D(\underline{x},s)-\log\tilde{D}(\underline{x},s)$ is absolutely convergent in
$\bar{H}_{x_{1}^{-1}}$ for each $\underline{x}\in\mathbb{R}_{+}^{l}$, hence a
continuous function $\bar{U}_{0}\to\mathbb{C}$ and a regular function
$\bar{H}_{x_{1}^{-1}}\to\mathbb{C}$ for each
$\underline{x}\in\mathbb{R}_{+}^{l}$. Let’s denote it by $h(\underline{x},s)$.
According to [7, Proposition 5.4], there exists some regular functions $g$
such that $g(s)\neq 0$ in $\overline{\mathbb{H}}_{1}$ and such that

$\prod_{p\in\mathcal{P}}(1+b(p)p^{-s})^{\phi(N)}=g(s)(s-1)^{\beta}.$

So, we have $\tilde{D}(\underline{x},s)=g(x_{1}s)(x_{i}s-1)^{\beta}$. Finally,

$D(\underline{x},s)^{\phi(N)}=\bigl{(}\tilde{D}(\underline{x},s)e^{h(\underline{x},s)}\bigr{)}^{\phi(N)}=e^{\phi(N)h(\underline{x},s)}g(x_{1}s)(x_{1}s-1)^{-\beta}.$

So we can just take $f(\underline{x},s)$ to be
$e^{h(\underline{x},s)}g(x_{1}s)x_{1}^{-\beta}$. ∎

######  Proposition 4.9.

Let $U=\mathbb{R}_{+}^{l}\times\mathbb{H}_{0}$ be an open subset of
$\mathbb{R}^{l}\times\mathbb{C}$. Let $\gamma>0$ be a positive integer, and
let $a_{i}:\mathcal{P}\times U\to\mathbb{C}$ be a coefficient with main term
$b$, where $0\neq b_{i}:\mathcal{P}\to\mathbb{R}_{+}$ is a non-negative
function modulo $N$, where $i=1,\dots,\gamma$. Define for each set
$I=\\{1=n_{1}\leq\cdots\leq n_{\gamma}\\}$ of positive integers the following
series

$D_{\gamma,I}(\underline{x},s):=\sum_{\begin{subarray}{c}\underline{p}\in\mathcal{P}^{\gamma}\\\
p_{1}\neq p_{2}\neq\cdots\neq
p_{\gamma}\end{subarray}}\prod_{i=1}^{\gamma}a_{i}(p,\underline{x},s)p_{i}^{-n_{i}x_{1}s}.$

Then $D_{\gamma,I}(\underline{x},s)$ defines a continuous function in a subset

$U_{0}:=\bigcup_{\underline{x}\in\mathbb{R}_{+}^{l}}\\{\underline{y}\in\mathbb{R}_{+}^{l}\mid
y_{1}\geq x_{1}\\}\times\mathbb{H}_{x_{1}^{-1}},$

of $U$, and for each $\underline{x}\in\mathbb{R}_{+}^{l}$, the function
$D_{\gamma,I}(\underline{x},s)$ is a regular function
$\mathbb{H}_{x_{1}^{-1}}\to\mathbb{C}$. Let

$\gamma^{\prime}=\sum_{j:n_{j}=1}1.$

Then there exists some continuous functions
$f_{0},\dots,f_{\gamma^{\prime}}:\bar{U}_{0}\to\mathbb{C}$ such that
$f_{\gamma^{\prime}}$ can be chosen to be a nonzero constant and such that for
each $\underline{x}\in\mathbb{R}_{+}^{l}$, we have

$D_{\gamma,I}(\underline{x},s)=\sum_{i=0}^{\gamma^{\prime}}f_{i}(\underline{x},s)\Bigl{(}\log\frac{1}{s-x_{1}^{-1}}\Bigr{)}^{i}.$

###### Proof.

For each $\underline{x}\in\mathbb{R}_{+}^{l}$, the series
$D_{\gamma,I}(\underline{x},s)$ is absolutely convergent if $\Re(x_{1}s)>1$,
i.e., when $s\in\mathbb{H}_{x_{1}^{-1}}$. So $D_{\gamma,I}$ defines a
continuous function $U_{0}\to\mathbb{C}$. Since each term in the series is
holomorphic with respect to $s$, we also see that
$D_{\gamma}(\underline{x},s)$ is a regular function in
$\mathbb{H}_{x_{1}^{-1}}$ for each $\underline{x}$.

If $\gamma=1$, then

$D_{1}(\underline{x},s)=\sum_{p\in\mathcal{P}}a_{1}(p,\underline{x},s)p^{-x_{1}s}.$

Define

$\tilde{D}_{1}(\underline{x},s):=\sum_{p\in\mathcal{P}}b_{1}(p)p^{-x_{1}s}.$

Similar argument also shows that $\tilde{D}_{1}(\underline{x},s)$ is a
continuous function in $U_{0}$, and for each
$\underline{x}\in\mathbb{R}_{+}^{l}$, it is a regular function in
$\mathbb{H}_{x_{1}^{-1}}$. Moreover, for each
$\underline{x}\in\mathbb{R}_{+}^{l}$ and for each
$s\in\mathbb{H}_{x_{1}^{-1}}$, we have

$\lvert
D_{1}(\underline{x},s)-\tilde{D}_{1}(\underline{x},s)\rvert\ll\sum_{n=1}^{\infty}p^{-(x_{1}+\delta_{1})\sigma},$

which shows that $h(\underline{x},s):=(D_{1}-\tilde{D}_{1})(\underline{x},s)$
is absolutely convergent in $\bar{H}_{x_{1}^{-1}}$ for each
$\underline{x}\in\mathbb{R}_{+}^{l}$. According to [7, Proposition 5.5], we
know that there exists some constant $g_{1}$ and some regular function $g_{0}$
in the closed half-plane $\bar{\mathbb{H}}_{1}$ such that

$\sum_{p\in\mathcal{P}}b_{1}(p)p^{-s}=g_{1}(s)\log\frac{1}{s-1}+g_{0}(s).$

So, we have

$D_{1}(\underline{x},s)=h(\underline{x},s)+g_{0}(x_{1}s)+g_{1}(x_{1}s)\log\frac{1}{x_{1}s-1},$

which concludes the proof when $\gamma=1$. Provided that the statement is true
for all $1,\dots,\gamma-1$. Let

$D_{1,a}(\underline{x},s):=\sum_{p\in\mathcal{P}}a(p,\underline{x},s)p^{-x_{1}s}.$

Then for a fixed set $I:=\\{1=n_{1}\leq\cdots\leq n_{\gamma}\\}$, let
$J:=\\{1=n_{1}\leq\cdots\leq n_{\gamma-1}\\}$, then

$\displaystyle
D_{\gamma-1,J}(\underline{x},s)D_{1,a_{\gamma}}(\underline{x},s)$
$\displaystyle=$ $\displaystyle
D_{\gamma,I}(\underline{x},s)+\sum_{i=1}^{\gamma-1}\sum_{\begin{subarray}{c}\underline{p}\in\mathcal{P}^{\gamma-1}\\\
p_{1}\neq\cdots\neq p_{\gamma-1}\end{subarray}}a_{1}p_{1}^{-n_{1}x_{1}s}\cdots
a_{i}a_{\gamma}p_{i}^{-(n_{i}+n_{\gamma})x_{1}s}\cdots
a_{\gamma-1}p_{\gamma-1}^{-n_{\gamma-1}x_{1}s}.$

Note that $a_{i}a_{\gamma}$ is also a coefficient function. For each
$i=1,\dots,\gamma-1$, the series

$\sum_{\begin{subarray}{c}\underline{p}\in\mathcal{P}^{\gamma-1}\\\
p_{1}\neq\cdots\neq p_{\gamma-1}\end{subarray}}a_{1}p_{1}^{-n_{1}x_{1}s}\cdots
a_{i}a_{\gamma}p_{i}^{-(n_{i}+n_{\gamma})x_{1}s}\cdots
a_{\gamma-1}p_{\gamma-1}^{-n_{\gamma-1}x_{1}s}$

satisfies the condition of our induction assumption by reordering
$n_{1},\dots,n_{i}+n_{\gamma},\dots,n_{\gamma-1}$ if necessary. So the
analytic continuation of $D_{\gamma,I}$ is proved by induction on $\gamma$. ∎

######  Remark.

It is clear that if $a=a_{1}=\cdots=a_{\gamma}$, and we define

$\tilde{D}_{\gamma,I}(s):=\sum_{\begin{subarray}{c}\underline{p}\in\mathcal{P}^{\gamma}\\\
p_{1}<p_{2}<\cdots<p_{\gamma}\end{subarray}}\prod_{i=1}^{\gamma}a(p,\underline{x},s)p_{i}^{-n_{i}x_{1}s},$

then
$\tilde{D}_{\gamma,I}(\underline{x},s)=\dfrac{1}{\gamma!}D_{\gamma,I}(\underline{x},s)$.

##  5\. Abelian extensions

Let $G$ be a finite abelian group. By viewing $G$ as a transitive permutation
group according to its action on itself, we see that
$\mathcal{S}:=\mathcal{S}(G)$ is just the set of abelian $G$-fields. In
Section 3, we’ve explained how to define an invariant of fields in
$\mathcal{S}$. If $\bar{G}=\\{\\{0\\},\Omega_{1},\dots,\Omega_{l}\\}$ is the
set of equivalence class under invertible powering, then Definition 3.2 gives
the definition $\vartheta_{\underline{x}}:\mathcal{S}\to\mathbb{R}_{+}$ of an
invariant with $l$ parameters $\underline{x}\in\mathbb{R}_{+}^{l}$. Let
$H_{i}:=\langle\Omega_{i}\rangle$ be the subgroup generated by $\Omega_{i}$,
for each $i=1,\dots,l$. It is clear that $H_{i}$ is a cyclic subgroup of $G$,
and $\\{\\{0\\},H_{1},\dots,H_{l}\\}$ is the set of all cyclic subgroups of
$G$. Recall also from Definition 3.7 that we have defined a function $R$ (or
the expression): given a subset $\Omega\subseteq G$ closed under invertible
powering, for each pair $(\gamma_{1},\gamma_{2})\in\mathbb{N}^{2}$, define

$R^{\gamma_{1}}_{\gamma_{2}}(\underline{x})=\lim_{x\to\infty}\frac{N(\mathcal{S},\vartheta_{\underline{x}};(\Omega,\gamma_{1});x)}{N(\mathcal{S},\vartheta_{\underline{x}};(\Omega,\gamma_{2});x)}.$

Let

$\bar{\Omega}=\\{\Lambda\cap\Omega\neq\emptyset\mid\Lambda\in\bar{G}\\}$

be the set of equivalence classes in $\Omega$. Assume without loss of
generality that

$\\{\Omega_{i}\in\bar{G}\mid\Omega_{i}\geq\Lambda\text{ for some
}\Lambda\in\bar{\Omega}\\}=\\{\Omega_{\xi_{1}},\dots,\Omega_{\xi_{u}}\\},$

where $\xi_{1}<\xi_{2}<\cdots<\xi_{u}$. If $\Omega=\emptyset$, then we make
the convention that $u=0$. By reordering the indices, we may assume without
loss of generality that $x_{1}\leq x_{2}\leq\cdots\leq x_{l}$, and let

$V:=\\{\underline{x}\in\mathbb{R}_{+}^{l}\mid x_{1}\leq\cdots\leq\cdots
x_{l}\\}.$

It is clear that the index set

$S(\underline{x}):=\\{d=e_{1}^{x_{1}}\cdots e_{l}^{x_{l}}\text{ where
}e_{1},\dots,e_{l}\in\mathbb{Z}_{+}\\}$

has parameter $\underline{x}\in\mathbb{R}_{+}^{l}$. We need a technical
notation to describe the difference between
$\operatorname{Hom}(\prod_{p}\mathbb{Z}_{p}^{*},G)$ and
$\operatorname{Sur}(\prod_{p}\mathbb{Z}_{p}^{*},G)$ when it comes to field
counting. For example, in the sense of asymptotic behaviour, there is no
difference between these two concept when $G\cong C_{q}$ where $q$ is a
rational prime. But in general, the result of counting non-surjective
homomorphisms may give a even larger main term than counting surjective ones.

######  Definition 5.1.

Let $\underline{x}\in V$ be a fixed parameter, and let $\Omega$ be a subset of
$G$ closed under invertible powering. For a homomorphism $\rho$, let
$\gamma(\rho)$ be the integer $\gamma$ such that
$\mathbf{1}_{(\Omega,\gamma)}(\rho)=1$. For each
$\rho\in\operatorname{Sur}(\prod_{p}\mathbb{Z}_{p}^{*},G)$, define

$\delta_{\underline{x}}(\rho):=\\#\\{p\nmid\lvert
G\rvert:\mathbf{1}_{\Omega}^{(p)}(\rho|_{p})=1\text{ and
}\vartheta_{\underline{x}}(\rho|_{p})>p^{x_{1}^{-1}}\\},$

Let
$\delta_{\underline{x}}:=\min\\{\delta_{\underline{x}}(\rho):\rho\in\operatorname{Sur}(\mathbb{Z}_{p}^{*},G)\\}$.
Define
$\gamma_{\underline{x}}:=\min\\{\gamma(\rho)\mid\rho\in\operatorname{Sur}(\prod_{p}\mathbb{Z}_{p}^{*},G)$
and $\delta_{\underline{x}}(\rho)=\delta_{\underline{x}}\\}$.

From the definition itself, one can show that if
$0<\gamma<\delta_{\underline{x}}$, then $\mathbf{1}_{(\Omega,\gamma)}(\rho)=0$
for all $\rho\in\operatorname{Sur}(\prod_{p}\mathbb{Z}_{p}^{*},G)$. Moreover,
we can see that actually $\delta_{\underline{x}}$, hence also
$\gamma_{\underline{x}}$, only depends on the set $\\{i\mid x_{i}>x_{1}\\}$.
The main theorem of this section can be stated as follows.

######  Theorem 5.2.

For a set of numbers $I=\\{1\leq n_{1}<n_{2}<\cdots<n_{m}\leq l\\}$, let
$V_{I}\subseteq V$ be the subset such that
$x_{1}=\cdots=x_{n_{1}}<x_{n_{1}+1}=\cdots=x_{n_{2}}<\cdots<x_{n_{m}+1}=\cdots=x_{l}$.
Let $\Omega\subseteq G$ be a nonempty subset closed under invertible powering.
Let $\gamma_{2}>\gamma_{1}\geq\gamma_{\underline{x}}$ be two integers. Then
$R^{\gamma_{1}}_{\gamma_{2}}(\underline{x})$ is a continuous function on
$V_{I}$. In addition, if $x_{\xi_{1}}>x_{1}$, then
$R^{\gamma_{1}}_{\gamma_{2}}(\underline{x})>0$. Else if $x_{\xi_{1}}=x_{1}$,
then $R^{\gamma_{1}}_{\gamma_{2}}(\underline{x})=0$.

We prove results on counting fields with parameter $\underline{x}$ to show the
above theorem. Inspired by Wood [8, Theorem 3.1], we make the following
definition.

######  Definition 5.3.

Let $G$ be an abelian group. For an element $g\in G$, we denote its order by
$r(g)$.

  1. (1)

Let $\Lambda$ be a subset of $G$ closed under invertible powering. Define

$\beta(\Lambda):=\sum_{\operatorname{id}\neq
g\in\Lambda}[\mathbb{Q}(\zeta_{r(g)}):\mathbb{Q}]^{-1}.$

In particular, let $x_{0}:=\min\\{x_{i}\mid i\neq\xi_{1},\dots,\xi_{u}\\}$,
then define

$\beta:=\sum_{\begin{subarray}{c}i\neq\xi_{1},\dots,\xi_{u}\\\
x_{i}=x_{0}\end{subarray}}\beta(\Omega_{i}).$

  2. (2)

Let $H$ be a subgroup. For each rational prime $p$, define

$\mathfrak{s}^{p}_{H}:=\lvert\operatorname{Sur}(\mathbb{Z}_{p}^{*},H)\rvert.$

######  Remark.

  1. (1)

One can check that the notation $\beta(\Lambda)$ is always an integer.

  2. (2)

Just like $\alpha$, the number $\beta$ is also a function
$\mathbb{R}^{l}\to\mathbb{R}$.

  3. (3)

If $H$ is cyclic and $p\nmid 2\lvert G\rvert$, then

$\operatorname{Sur}(\mathbb{Z}_{p}^{*},H)\leftrightarrow\\{h\in H\mid p\equiv
1\mod r(h)\\},$

where $\leftrightarrow$ means that there exists a 1-1 correspondence. Because
in this case, the image $\rho_{p}(\mathbb{Z}_{p}^{*})$ is totally determined
by the image of a generator $\xi_{p-1}$ of the group of roots of unity. So,
$\rho_{p}(\mathbb{Z}_{p}^{*})=H$ if and only if the image
$\rho_{p}(\xi_{p-1})=h$ has order divisible by $p-1$, i.e., $p-1\equiv 0\bmod
r(h)$.

######  Lemma 5.4.

Let $d\in S(\underline{x})$ be an index. If $d=p_{1}^{x(F_{1})}\cdots
p_{n}^{x(F_{n})}$ where $p_{1},\dots,p_{n}$ are rational primes and
$F_{1},\dots,F_{n}$ are subgroups of $G$, then there are exactly

$\prod_{j=1}^{n}\mathfrak{s}^{p_{j}}_{F_{j}}$

different homomorphisms $\rho:\prod_{p}\mathbb{Z}_{p}^{*}\to G$ such that
$\vartheta_{\underline{x}}(\rho)=d$ and such that
$\operatorname{im}(\rho|_{p_{j}})=F_{j}$ for all $j=1,\dots,n$.

###### Proof.

Recall from Definition 3.2 and 3.3 that $\vartheta_{\underline{x}}$ and
$\mathbf{1}_{(\Omega,\gamma)}$ are both defined locally. So we see that
$\operatorname{Hom}(\prod_{p}\mathbb{Z}_{p}^{*},G)$ is equipped with an
invariant $\vartheta_{\underline{x}}$. To be precise, for any rational prime
$p$, let $\rho_{p}:\mathbb{Z}_{p}^{*}\to G$ be a local morphism. If the image
$\operatorname{im}(\rho_{p})=H$ where $H$ is a subgroup, then
$\vartheta_{\underline{x}}(\rho_{p})=x(H)$ according to our definition of
$\vartheta_{\underline{x}}$.

By a particular decomposition $d=p_{1}^{x(F_{1})}\cdots p_{n}^{x(F_{n})}$, we
actually specify the image of $\rho_{p}$ for each $p$. According to the
definition of $\mathfrak{s}^{p}_{H}$, we see that for each pair $(p,H)$, there
are exactly $\mathfrak{s}^{p}_{H}$ local homomorphisms
$\rho_{p}:\mathbb{Z}_{p}^{*}\to G$ such that $\operatorname{im}(\rho_{p})=H$.
Since a homomorphism $\rho$ can be obtained by taking the product of local
homomorphisms, i.e., $\rho=\prod_{p}\rho_{p}$, we see that the statement is
true. ∎

Recall that for an integer $N$, we’ve defined the notation
$\mathcal{P}(N)=\\{p\in\mathcal{P}\mid p\nmid N\\}$. If $\Omega\neq\emptyset$,
we define for each $\underline{x}\in V$ and each natural number
$\gamma=0,1,\dots$, the following series

$\displaystyle\mu_{0}(\underline{x},s):$ $\displaystyle=\sum_{p\mid\lvert
G\rvert}(1+\sum_{\begin{subarray}{c}H\subseteq G\\\
H\cap\Omega=\emptyset\end{subarray}}\mathfrak{s}^{p}_{H}p^{-x(H)s})\cdot\prod_{p\in\mathcal{P}(\lvert
G\rvert)}(1+\sum_{\begin{subarray}{c}H\subseteq G\\\
H\cap\Omega=\emptyset\end{subarray}}\mathfrak{s}^{p}_{H}p^{-x(H)s})$
$\displaystyle\mu_{\gamma}(\underline{x},s):$
$\displaystyle=\sum_{\begin{subarray}{c}\underline{p}\in\mathcal{P}(\lvert
G\rvert)^{\gamma}\\\
p_{1}<\cdots<p_{\gamma}\end{subarray}}\prod_{j=1}^{\gamma}(\sum_{\begin{subarray}{c}H\subseteq
G\\\
H\cap\Omega\neq\emptyset\end{subarray}}\mathfrak{s}^{p_{j}}_{H}p_{j}^{-x(H)s})$
$\displaystyle\cdot\prod_{\begin{subarray}{c}p\in\mathcal{P}(\lvert
G\rvert)\\\ p\nmid p_{1}\cdots
p_{\gamma}\end{subarray}}(1+\sum_{\begin{subarray}{c}H\subseteq G\\\
H\cap\Omega=\emptyset\end{subarray}}\mathfrak{s}^{p}_{H}p^{-x(H)s})\cdot\prod_{p\mid\lvert
G\rvert}(1+\sum_{H\subseteq G}\mathfrak{s}^{p}_{H}p^{-x(H)s})$

And let

$\pi_{\gamma}(\underline{x},s):=\sum_{\begin{subarray}{c}d\in
S(\underline{x})\\\
\vartheta_{\underline{x}}(K)=d\end{subarray}}\mathbf{1}_{(\Omega,\gamma)}(K).$

When $\Omega=\emptyset$, we define

$\pi(\underline{x},s)=\sum_{\begin{subarray}{c}d\in S(\underline{x})\\\
\vartheta_{\underline{x}}(K)=d\end{subarray}}1\quad\text{and}\quad\mu(\underline{x},s)=\prod_{p\in\mathcal{P}}(1+\sum_{H\subseteq
G}\mathfrak{s}^{p}_{H}p^{-x(H)s}).$

In this case, by abuse of notation, we write
$\mu_{\gamma}(\underline{x},s)=\mu(s)$ and
$\pi_{\gamma}(\underline{x},s)=\pi(s)$ for all $\gamma=0,1,2,\dots$.

######  Lemma 5.5.

For each $\underline{x}\in V$, if we write

$\mu_{\gamma}(\underline{x},s)=\sum_{d\in
S(\underline{x})}a_{d}d^{-s}\quad\text{and}\quad\pi_{\gamma}(s)=\sum_{d\in
S}b_{d}d^{-s}$

in the realm of absolute convergence, then

$\displaystyle
N(\operatorname{Hom}(\prod_{p}\mathbb{Z}_{p}^{*},G),\vartheta_{\underline{x}};(\Omega,\gamma);x)=\sum_{d<x}a_{d}$
$\displaystyle
N(\operatorname{Sur}(\prod_{p}\mathbb{Z}_{p}^{*},G),\vartheta_{\underline{x}};(\Omega,\gamma);x)=\sum_{d<x}b_{d}.$

In addition, $\mu_{\gamma}(\underline{x},s)\geq\pi_{\gamma}(\underline{x},s)$,
i.e., $a_{d}\geq b_{d}$ for all $d\in S$.

###### Proof.

In Lemma 5.4, we’ve shown that
$\operatorname{Hom}(\prod_{p}\mathbb{Z}_{p}^{*},G)$ is equipped with an
invariant $\vartheta_{\underline{x}}$. Moreover, when $p\nmid\lvert G\rvert$,
we have $H\cap\Omega\neq\emptyset$ if and only if
$\mathbf{1}_{\Omega}^{(p)}(\rho_{p})=1$. So
$\operatorname{Hom}(\prod_{p}\mathbb{Z}_{p}^{*},G)$ is also equipped with
$\mathbf{1}_{(\Omega,\gamma)}$. If $\tilde{\rho}:\mathbf{C}_{\mathbb{Q}}\to G$
is an Artin reciprocity map, then let $\rho:=\prod_{p}\tilde{\rho}|_{p}$. We
see that
$\vartheta_{\underline{x}}(\tilde{\rho})=\vartheta_{\underline{x}}(\rho)=\prod_{p}\vartheta_{\underline{x}}(\rho_{p})$,
and
$\mathbf{1}_{(\Omega,\gamma)}(\tilde{\rho})=\mathbf{1}_{(\Omega,\gamma)}(\rho)$.

For each $d\in S$, we can decompose it into $d=p_{1}^{x(F_{1})}\cdots
p_{n}^{x(F_{n})}$. Warning: the decomposition may not be unique! For example
the case when $\vartheta_{\underline{x}}$ is the product of ramified primes,
i.e., $x_{0}=\cdots=x_{l}=1$. According to Lemma 5.4, we see that, if
$\Omega=\emptyset$, then

$\sum_{\vartheta_{\underline{x}}(\rho)=d}1=\sum_{\begin{subarray}{c}d=p_{1}^{x(F_{1})}\cdots
p_{n}^{x(F_{n})}\\\ p_{i}\in\mathcal{P},\,F_{i}\subseteq
G\end{subarray}}\prod_{i=1}^{n}\mathfrak{s}^{p_{j}}_{F_{j}}=a_{d},$

where the sum is taken over all possible decomposition
$d=p_{1}^{x(F_{1})}\cdots p_{n}^{x(F_{n})}$. Else if $\Omega\neq\emptyset$,
then we only take into account the decomposition $d=p_{1}^{x(F_{1})}\cdots
p_{n}^{x(F_{n})}$ satisfying the condition that there are exactly $\gamma$
pairs $(p_{j},F_{j})$ such that $p_{j}\nmid\lvert G\rvert$ and such that the
subgroup $F_{j}=H_{i}$ for some $i=0,\dots,m$, then we get

$\sum_{\vartheta_{\underline{x}}(\rho)=d}\mathbf{1}_{(\Omega,\gamma)}(\rho)=\sum_{d=p_{1}^{x(F_{1})}\cdots
p_{n}^{x(F_{n})}}\prod_{i=j}^{n}\mathfrak{s}^{p_{j}}_{F_{j}}=a_{d}.$

Since, according to Class Field Theory, we know that

$\operatorname{Sur}(\mathbf{C}_{\mathbb{Q}},G)=\operatorname{Sur}(\prod_{p}\mathbb{Z}_{p}^{*},G).$

If $\rho\in\operatorname{Sur}(\prod_{p}\mathbb{Z}_{p}^{*},G)$ is a surjective
homomorphism, then it corresponds to a class field $K\in\mathcal{S}$ and vice
versa. This gives the second identity in the statement. Since each surjective
homomorphism $\rho:\prod_{p}\mathbb{Z}_{p}^{*}\to G$ is already a
homomorphism, we see that $a_{d}\geq b_{d}$ for all $d\in S$. ∎

Then we can state the result on the analytic continuation of
$\pi_{\gamma}(s)$.

######  Theorem 5.6.

Let $I=\\{1\leq n_{1}<n_{2}<\cdots<n_{m}\leq l\\}$ be a set of numbers, and
let $V_{I}\subseteq V$ be the subset such that $x_{1}=x_{2}=\cdots
x_{n_{1}},x_{n_{1}+1}=\cdots=x_{n_{2}},\dots,x_{n_{m}+1}=\cdots=x_{l}$. For
each natural number $\gamma$, the expression $\pi_{\gamma}(\underline{x},s)$
defines a continuous function in a subset

$U_{I}:=\bigcup_{\underline{x}\in V_{I}}\\{\underline{y}\in V_{I}\mid
y_{1}\geq x_{1}\\}\times\mathbb{H}_{x_{1}^{-1}}$

of $V_{I}\times\mathbb{H}_{0}$. For each $\underline{x}\in V_{I}$, the
function $\pi_{\gamma}(\underline{x},s)$ is absolutely convergent and regular
in the open half-plane $\sigma>x_{1}^{-1}$. Assume in addition that
$\gamma\geq\gamma_{\underline{x}}$. If $\Omega=\emptyset$ or $x_{1}>x_{2}$,
then there exists continuous functions
$f(\underline{x},s),f_{0}(\underline{x},s):\bar{U}_{I}\to\mathbb{C}$ such that
for each $\underline{x}\in V_{I}$, $f,f_{0}$ are regular in the closed half-
plane $\sigma\geq x_{1}^{-1}$ with $f_{0}(\underline{x},x_{1}^{-1})\neq 0$,
and

$\pi_{\gamma}(\underline{x},s)=f(\underline{x},s)+f_{0}(\underline{x},s)(s-x_{1}^{-1})^{\beta}.$

Else if $\Omega\neq\emptyset$ and $x_{i_{1}}=x_{1}$, then there exists
continuous functions
$f(\underline{x},s),f_{0}(\underline{x},s),\dots,f_{\gamma-\delta_{\underline{x}}}(\underline{x},s):\bar{U}_{I}\to\mathbb{C}$
such that for each $\underline{x}\in V_{I}$,
$f,f_{0},\dots,f_{\gamma-\delta_{\underline{x}}}$ are regular in the closed
half-plane $\sigma\geq x_{1}^{-1}$ such that
$f_{\gamma-\delta_{\underline{x}}}(\underline{x},x_{1}^{-1})\neq 0$ and that

$\pi_{\gamma}(s)=f(s)+(s-x_{0}^{-1})^{\beta}\sum_{j=0}^{\gamma-\delta_{\underline{x}}}f_{j}(\underline{x},s)\big{(}\log\frac{1}{s-x_{i_{1}}}\big{)}^{j},$

where $x_{0}=\min\\{x_{i}\mid i\neq\xi_{1},\dots,\xi_{u}\\}$.

Using the above statement, we can first prove the main results of this
section.

###### Proof of Theorem 5.2.

Let $\Omega$ be a nonempty subset of $G$ that is closed under invertible
powering. If $x_{\xi_{1}}>x_{1}$, according to Theorem 5.6, there exists some
functions $f_{\gamma_{1}},f_{\gamma_{1},0},f_{\gamma_{2}},f_{\gamma_{2},0}$ in
$\bar{U}_{I}$ such that

$\pi_{\gamma_{i}}(\underline{x},s)=f_{\gamma_{i}}(\underline{x},s)+f_{\gamma_{i},0}(\underline{x},s)(s-x_{1}^{-1})^{\beta}$

where $i=1,2$. Then by Theorem 4.1, we have

$\displaystyle R^{\gamma_{1}}_{\gamma_{2}}(\underline{x})=$
$\displaystyle\lim_{x\to\infty}\frac{N(\mathcal{S},\vartheta(\underline{x});(\Omega,\gamma_{1});x)}{N(\mathcal{S},\vartheta(\underline{x});(\Omega,\gamma_{2});x)}$
$\displaystyle=$
$\displaystyle\lim_{x\to\infty}\frac{(f_{\gamma_{1},0}(\underline{x},x_{1}^{-1})\Gamma(\beta)^{-1}+o(1))x^{x_{1}^{-1}}(\log
x)^{\beta-1}}{(f_{\gamma_{2},0}(\underline{x},x_{1}^{-1})\Gamma(\beta)^{-1}+o(1))x^{x_{1}^{-1}}(\log
x)^{\beta-1}}$ $\displaystyle=$
$\displaystyle\frac{f_{\gamma_{1},0}(\underline{x},x_{1}^{-1})}{f_{\gamma_{2},0}(\underline{x},x_{1}^{-1})}$

We can see that $R^{\gamma_{1}}_{\gamma_{2}}(\underline{x})>0$ because
$f_{\gamma_{i},0}(\underline{x},x_{1}^{-1})>0$ for each $\underline{x}\in
V_{I}$. Moreover, $f_{\gamma_{i},0}(\underline{x},s):\bar{U}_{I}\to\mathbb{C}$
is a continuous function, so $f_{\gamma_{i},0}(\underline{x},x_{1}^{-1})$ is a
continuous function on $V_{I}$, hence also
$R^{\gamma_{1}}_{\gamma_{2}}(\underline{x})$. Similar argument works for the
case when $x_{\xi_{1}}=x_{1}$ except that
$R^{\gamma_{1}}_{\gamma_{2}}(\underline{x})=0$ for all $\underline{x}\in
V_{I}$ in this case. ∎

We prove the statement of Theorem 5.6 step by step. It suffices to prove the
theorem for a fixed set $I=\\{1\leq n_{1}<n_{2}<\cdots<n_{m}\leq l\\}$ of
numbers. Let $V_{I}\subseteq V$ be the subset such that $x_{1}=x_{2}=\cdots
x_{n_{1}},x_{n_{1}+1}=\cdots=x_{n_{2}},\dots,x_{n_{m}+1}=\cdots=x_{l}$, and
let

$U_{I}:=\bigcup_{\underline{x}\in V_{I}}\\{\underline{y}\in V_{I}\mid
y_{1}\geq x_{1}\\}\times\mathbb{H}_{x_{1}^{-1}}$

be a subset of $V_{I}\times\mathbb{H}_{0}$. The method we adopt applies to all
the cases, so in this section we just assume that $\Omega\neq\emptyset$ and
$\gamma>1$. In this case, first recall that

$\displaystyle\mu_{\gamma}(\underline{x},s)=$
$\displaystyle\sum_{\begin{subarray}{c}\underline{p}\in\mathcal{P}(\lvert
G\rvert)^{\gamma}\\\
p_{1}<\cdots<p_{\gamma}\end{subarray}}\prod_{j=1}^{\gamma}(\sum_{k=1}^{u}\mathfrak{s}^{p_{j}}_{H_{\xi_{k}}}p_{j}^{-x_{\xi_{k}}s})\cdot$
$\displaystyle\prod_{p\mid\lvert G\rvert}(1+\sum_{H\subseteq
G}\mathfrak{s}^{p}_{H}p^{-x(H)s})\prod_{\begin{subarray}{c}p\in\mathcal{P}(\lvert
G\rvert)\\\ p\nmid p_{1}\cdots
p_{\gamma}\end{subarray}}(1+\sum_{\begin{subarray}{c}1\leq i\leq l\\\
i\neq\xi_{1},\dots,\xi_{u}\end{subarray}}\mathfrak{s}^{p}_{H_{i}}p^{-x_{i}s})$

Let’s first prove the analytic continuation of $\mu_{\gamma}(\underline{x},s)$
in this case.

######  Lemma 5.7.

The expression $\mu_{\gamma}(\underline{x},s)$ defines a continuous function
in $U_{I}$ that is a regular function in the open half-plane
$\sigma>x_{1}^{-1}$ for each $\underline{x}\in V_{I}$. There exists continuous
functions
$f(\underline{x},s),g_{0}(\underline{x},s),\dots,g_{\gamma}(\underline{x},s)$
in $\bar{U}_{I}$ that are regular in the closed half-plane $\sigma\geq
x_{1}^{-1}$ for each $\underline{x}\in\mathbb{R}_{+}^{l+1}$, such that
$g_{\gamma}(\underline{x},x_{\xi_{1}}^{-1})$ is some nonzero constant and
$f(\underline{x},x_{0}^{-1})\neq 0$, and such that

$\mu_{\gamma}(\underline{x},s)=f(\underline{x},s)(s-x_{0}^{-1})^{\beta}\sum_{j=0}^{\gamma}g_{j}(\underline{x},s)\big{(}\log\frac{1}{s-x_{\xi_{1}}^{-1}}\big{)}^{j}.$

###### Proof.

Let’s first rewrite the series. By comparing with a large enough power of
Riemann-zeta function, we see that for each $\underline{x}$, the expression
$\mu_{\gamma}(\underline{x},x_{1}s)$ is absolutely convergent in $\sigma>1$,
hence $\mu_{\gamma}(\underline{x},s)$ is a regular function in the open half-
plane $\sigma>x_{1}^{-1}$. We first write

$\sum_{\begin{subarray}{c}1\leq i\leq l\\\
i\neq\xi_{1},\dots,\xi_{u}\end{subarray}}\mathfrak{s}^{p}_{H_{i}}p^{-x_{i}s}=a(p,\underline{x},s)p^{-x_{0}s},$

where $a$ is a coefficient function of main term

$\sum_{\begin{subarray}{c}1\leq i\leq l\\\
i\neq\xi_{1},\dots,\xi_{u}\end{subarray}}\mathfrak{s}^{p}_{H_{i}}.$

Then in $U_{I}$, we have

$\displaystyle\mu_{\gamma}(\underline{x},s)=$ $\displaystyle\prod_{p\mid\lvert
G\rvert}(1+\sum_{H\subseteq
G}\mathfrak{s}^{p}_{H}p^{-x(H)s})\prod_{p\in\mathcal{P}(\lvert
G\rvert)}(1+a(p,\underline{x},s)p^{-x_{0}s})$ $\displaystyle\cdot$
$\displaystyle\prod_{\begin{subarray}{c}\underline{p}\in\mathcal{P}^{\gamma}(\lvert
G\rvert)\\\
p_{1}<\cdots<p_{\gamma}\end{subarray}}\prod_{j=1}^{\gamma}\Bigl{(}\sum_{k=1}^{u}\frac{\mathfrak{s}^{p_{j}}_{H_{\xi_{k}}}}{1+a(p_{j},\underline{x},s)p_{j}^{-x_{0}s}}p_{j}^{-x_{\xi_{k}}s}\Bigr{)}.$

Since for each $p\nmid\lvert G\rvert$, the expression

$\sum_{k=1}^{u}\frac{\mathfrak{s}^{p}_{H_{\xi_{k}}}}{1+a(p,\underline{x},s)p^{-x_{0}s}}p^{-x_{\xi_{k}}s}$

is a coefficient function $b(p,\underline{x},s)$ of main term

$\sum_{k:x_{\xi_{k}}=x_{\xi_{1}}}\mathfrak{s}^{p}_{H_{\xi_{k}}}.$

By the projection map $x_{i}\mapsto y_{j}$ where $n_{j}\leq i\leq n_{j+1}$ and
$j=1,\dots,m$, and Proposition 4.9, we see that there exists
$g_{0},\dots,g_{\gamma}$ satisfying the conditions in the statement such that

$\prod_{\begin{subarray}{c}\underline{p}\in\mathcal{P}^{\gamma}(\lvert
G\rvert)\\\
p_{1}<\cdots<p_{\gamma}\end{subarray}}\prod_{j=1}^{\gamma}b(p_{j},\underline{x},s)p_{j}^{-x_{\xi_{1}}s}=\sum_{j=1}^{\gamma}g_{j}(\underline{x},s)\Bigl{(}\log\frac{1}{s-x_{\xi_{1}}^{-1}}\Bigr{)}^{\gamma}.$

Then Proposition 4.8 finishes the proof except for the formula of analytic
continuation. Proposition 4.8 says that

$\prod_{p\in\mathcal{P}(\lvert
G\rvert)}(1+a(p,\underline{x},s)p^{-x_{0}s})^{\phi(\lvert
G\rvert)}=g(\underline{x},s)(s-x_{1}^{-1})^{\beta^{\prime}},$

where $g$ is a continuous function in $\bar{U}_{I}$ and for each
$\underline{x}\in V_{I}$ the function $g(\underline{x},s)$ is regular in the
closed half-plane $\sigma\geq x_{1}^{-1}$ and non-zero everywhere, and

$\displaystyle\beta^{\prime}=$
$\displaystyle\sum_{\begin{subarray}{c}n\in(\mathbb{Z}/\lvert G\rvert)^{*}\\\
p\equiv n\bmod{\lvert
G\rvert}\end{subarray}}\sum_{i=1}^{n_{1}}\mathfrak{s}^{p}_{H_{i}}=\sum_{i=1}^{n_{1}}\sum_{\begin{subarray}{c}n\in(\mathbb{Z}/\lvert
G\rvert)^{*}\\\ p\equiv n\bmod{\lvert
G\rvert}\end{subarray}}\mathfrak{s}^{p}_{H_{i}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{1}}\frac{\phi(\lvert G\rvert)}{\phi(\lvert
H_{i}\rvert)}\cdot\phi(\lvert H_{i}\rvert)=\phi(\lvert G\rvert)n_{1}.$

On the other hand, for each $i=1,\dots,l$, we have

$\beta(\Omega_{i})=\sum_{g\in\Omega_{i}}[\mathbb{Q}(\zeta_{r(g)}):\mathbb{Q}]^{-1}=\sum_{g\in\Omega_{i}}\frac{1}{\phi(H_{i})}=1.$

Therefore $\beta=\sum_{i=1}^{n_{1}}1=n_{1}$. So there exists $f$ satisfying
the conditions in the statement such that

$\prod_{p\mid\lvert G\rvert}(1+\sum_{H\subseteq
G}\mathfrak{s}^{p}_{H}p^{-x(H)s})\prod_{p\in\mathcal{P}(\lvert
G\rvert)}(1+a(p,\underline{x},s)p^{-x_{0}s})=f(\underline{x},s)(s-x_{0}^{-1})^{-\beta}.$

And we are done for the proof. ∎

Then we construct some generating series to estimate the bounds of field
counting. Let’s introduce some notations.

######  Definition 5.8.

Let $\gamma\geq\gamma_{\underline{x}}$ be a fixed integer. If
$\gamma=n_{\xi_{1}}+\cdots+n_{\xi_{u}}$ is a partition into non-negative
integers such that there exists some homomorphism $\rho_{0}:\prod_{p\mid\lvert
G\rvert}\mathbb{Z}_{p}^{*}\to G$ and such that

$\operatorname{im}(\rho_{0})\cup\bigcup_{\begin{subarray}{c}1\leq i\leq l\\\
i\neq\xi_{1},\dots,\xi_{u}\end{subarray}}H_{i}\cup\bigcup_{k:n_{\xi_{k}}>0}H_{\xi_{k}}$

can generate the whole group $G$, then we say that the partition is
_admissible_. On the other hand, let $\rho:\prod_{p}\mathbb{Z}_{p}^{*},G$ be a
homomorphism. Define

$n_{\xi_{k}}(\rho):=\\#\\{p\in\mathcal{P}(\lvert
G\rvert):\operatorname{im}(\rho|_{p})=H_{\xi_{k}}\\},$

for all $k=1,\dots,u$.

It is clear that if $\rho$ is surjective and $\gamma(\rho)=\gamma$, then

$\gamma=\sum_{k=1}^{u}n_{\xi_{k}}(\rho)$

is an admissible partition, because we can just take $\rho_{0}$ to be the
restriction of $\rho$ at the places $\\{p\mid\lvert G\rvert\\}$.

######  Lemma 5.9.

If $x_{\xi_{1}}=x_{0}$ and $\gamma\geq\gamma_{\underline{x}}$, then there
exists some $\rho\in\operatorname{Sur}(\prod_{p\mid\lvert
G\rvert}\mathbb{Z}_{p}^{*},G)$ such that
$\delta_{\underline{x}}(\rho)=\delta_{\underline{x}}$ and such that
$\gamma(\rho)=\gamma$.

###### Proof.

According to Definition 5.1, there exists some
$\rho^{\prime}\in\operatorname{Sur}(\prod_{p\mid\lvert
G\rvert}\mathbb{Z}_{p}^{*},G)$ such that
$\delta_{\underline{x}}(\rho^{\prime})=\delta_{\underline{x}}$ and such that
$\gamma(\rho^{\prime})=\gamma_{\underline{x}}$. If
$\gamma=\gamma_{\underline{x}}$, then we are done. Otherwise, write
$\gamma^{\prime}:=\gamma-\gamma_{\underline{x}}$, which is a positive integer,
and let $p_{1},\dots,p_{\gamma^{\prime}}$ be some rational primes such that
$p_{i}\equiv 1\bmod{\lvert G\rvert}$ and such that $\rho^{\prime}|_{p_{i}}=1$,
for all $i=1,\dots,\gamma^{\prime}$. For each $i$, we can pick up a surjective
local homomorphism $\rho_{p_{i}}:\mathbb{Z}_{p_{i}}^{*}\to H_{\xi_{1}}$. Then
define $\rho:=\rho^{\prime}\cdot\prod_{i=1}^{\gamma^{\prime}}\rho_{p_{i}}$. It
is clear that $\mathbf{1}_{(\Omega,\gamma)}(\rho)=1$ and $\rho$ is surjective.
Moreover, if a rational prime $p$ satisfies the condition that
$\operatorname{im}(\rho|_{p})=H_{\xi_{k}}$ for some $k>1$, then $p\neq p_{i}$
for all $i$. So, we know that
$\delta_{\underline{x}}(\rho)=\delta_{\underline{x}}(\rho^{\prime})=\delta_{\underline{x}}$.
∎

We need to construct another function $\psi_{\gamma}(s)$ such that
$\psi_{\gamma}(s)\geq\pi_{\gamma}(s)$. Because $\mu_{\gamma}(s)$ gives too
large an estimate for
$N(\operatorname{Sur}(\prod_{p}\mathbb{Z}_{p}^{*},G),\vartheta_{\underline{x}};(\Omega,\gamma);x)$
when $x_{i_{1}}\leq x_{1}$.

######  Lemma 5.10.

Let $\gamma\geq\gamma_{\underline{x}}$ be a fixed integer. For each
$\underline{x}\in V_{I}$, there exists a generating series
$\psi_{\gamma}(\underline{x},s)=\sum_{d}a_{d}d^{-s}$ such that
$\psi_{\gamma}(\underline{x},s)\geq\pi_{\gamma}(\underline{x},s)$ and such
that it defines a regular function in the open half-plane $\sigma>x_{1}^{-1}$
which admits the following analytic continuation. there exists regular
functions $f,g_{0},\dots,g_{\gamma-\delta_{\underline{x}}}$ such that
$g_{\gamma-\delta_{\underline{x}}}$ can be chosen to be a non-zero constant
and $f(x_{0})\neq 0$, and such that

$\psi_{\gamma}(\underline{x},s)=f(s)(s-x_{0}^{-1})^{-\beta}\sum_{j=0}^{\gamma-\delta_{\underline{x}}}g_{j}(s)\big{(}\log\frac{1}{s-x_{\xi_{1}}^{-1}}\big{)}^{j}.$

###### Proof.

Let $\omega_{m}(n)=\\#\\{p\nmid m:p\mid n\\}$. Let’s construct a generating
series as follows.

$\displaystyle\psi_{\gamma}(\underline{x},s)=$
$\displaystyle\prod_{p\mid\lvert G\rvert}(1+\sum_{H\subseteq
G}\mathfrak{s}^{p}_{H}p^{-x(H)s})\prod_{p\in\mathcal{P}(\lvert
G\rvert)}(1+\sum_{\begin{subarray}{c}1\leq i\leq l\\\
i\neq\xi_{1},\dots,\xi_{u}\end{subarray}}\mathfrak{s}^{p}_{H_{i}}p^{-x_{i}s})$
$\displaystyle\sum_{\begin{subarray}{c}\gamma=n_{\xi_{1}}+\cdots+n_{\xi_{u}}\\\
\text{admissible
partition}\end{subarray}}\prod_{k=1}^{u}\prod_{\begin{subarray}{c}\underline{p}\in\mathcal{P}(\lvert
G\rvert)^{n_{\xi_{k}}}\\\
p_{1}<\cdots<p_{n_{\xi_{k}}}\end{subarray}}\mathfrak{s}^{p_{1}}_{H_{\xi_{k}}}p_{1}^{-x_{\xi_{k}}s}\cdots\mathfrak{s}^{p_{n_{\xi_{k}}}}_{H_{\xi_{k}}}p_{n_{\xi_{k}}}^{-x_{\xi_{k}}s}.$

The expression $\psi_{\gamma}(\underline{x},s)$ defines a continuous function
in $U_{I}$ that is a regular function in the open half-plane $\sigma>x_{1}$
for each $\underline{x}\in V_{I}$, whose proof is similar to that of
$\mu_{\gamma}(\underline{x},s)$. So, we can write it as
$\psi_{\gamma}(\underline{x},s)=\sum_{d}a_{d}d^{-s}$ when $\underline{x}\in
V_{I}$. If $\rho\in\operatorname{Sur}(\prod_{p}\mathbb{Z}_{p}^{*},G)$ is a
surjective homomorphism such that $\gamma(\rho)=\gamma$, then $\rho$ can give
an admissible partition as

$\gamma=\sum_{k=1}^{u}n_{\xi_{k}}(\rho).$

If a homomorphism $\rho$ gives an admissible partition via
$\gamma=\sum_{k}n_{\xi_{k}}(\rho)$, then we call it admissible with respect to
$\gamma$. Note that if $\rho$ is admissible with respect to $\gamma$, then
$\mathbf{1}_{(\Omega,\gamma)}(\rho)=1$. So, using Lemma 5.4, we see that

$\displaystyle a_{d}=$ $\displaystyle\sum_{d=e_{1}^{x_{1}}\cdots
e_{l}^{x_{l}}}\prod_{i=1}^{l}\prod_{p\mid e_{i}}\mathfrak{s}^{p}_{H_{i}}$
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}\rho\text{
admissible}\\\
\vartheta_{\underline{x}}(\rho)=d\end{subarray}}1\geq\sum_{\begin{subarray}{c}\rho\in\operatorname{Sur}(\prod_{p}\mathbb{Z}_{p}^{*},G)\\\
\vartheta_{\underline{x}}(\rho)=d\end{subarray}}\mathbf{1}_{(\Omega,\gamma)}(\rho),$

where $e_{1},\dots,e_{m}$ are square-free integers such that
$\gamma=\omega_{\lvert G\rvert}(e_{\xi_{1}})+\cdots+\omega_{\lvert
G\rvert}(e_{\xi_{u}})$ is admissible. This implies that

$\psi_{\gamma}(\underline{x},s)\geq\pi_{\gamma}(\underline{x},s),$

as required. The proof of the analytic properties of $\psi$ is similar to that
of $\mu$. We only show the part of analytic continuation here. Let $\rho$ be a
surjective homomorphism such that $\gamma(\rho)=\gamma$. Let
$\gamma=\sum_{k}n_{\xi_{k}}$ be the corresponding partition. Note that in this
case

$\sum_{\begin{subarray}{c}1\leq k\leq u\\\
x_{\xi_{k}}=x_{\xi_{1}}\end{subarray}}n_{\xi_{k}}=\gamma-\delta_{\underline{x}}(\rho).$

Then according to Proposition 4.8 and 4.9, there exists continuous functions
$f_{\rho},g_{\rho,0},\dots,g_{\rho,\gamma-\delta_{\underline{x}}(\rho)}:\bar{U}_{I}\to\mathbb{C}$
that are regular in the closed half-plane $\sigma\geq x_{1}^{-1}$ for each
$\underline{x}\in V_{I}$ such that
$g_{\rho,\gamma-\delta_{\underline{x}}(\rho)}$ can be chosen to be a positive
constant and $f_{\rho}(\underline{x},x_{0})\neq 0$ and such that

$\displaystyle\prod_{p\in\mathcal{P}(\lvert
G\rvert)}(1+\sum_{\begin{subarray}{c}1\leq i\leq l\\\
i\neq\xi_{1},\dots,\xi_{u}\end{subarray}}\mathfrak{s}^{p}_{H_{i}}p^{-x_{i}s})\prod_{k=1}^{u}\prod_{\begin{subarray}{c}\underline{p}\in\mathcal{P}^{n_{\xi_{k}}}(\lvert
G\rvert)\\\
p_{1}<\cdots<p_{n_{\xi_{k}}}\end{subarray}}\mathfrak{s}^{p_{1}}_{H_{\xi_{k}}}p_{1}^{-x_{\xi_{k}}s}\cdots\mathfrak{s}^{p_{n_{\xi_{k}}}}_{H_{\xi_{k}}}p_{n_{\xi_{k}}}^{-x_{\xi_{k}}s}$
$\displaystyle=$ $\displaystyle
f(\underline{x},s)(s-x_{0}^{-1})^{-\beta}\sum_{j=0}^{\gamma-\delta_{\underline{x}}(\rho)}g_{j}(\underline{x},s)\big{(}\log\frac{1}{s-x_{\xi_{1}}^{-1}}\big{)}^{j}.$

Since $\gamma\geq\gamma_{\underline{x}}$, there exists at least one surjective
homomorphism $\rho$ such that
$\delta_{\underline{x}}(\rho)=\delta_{\underline{x}}$ and
$\gamma(\rho)=\gamma_{\underline{x}}$. By summing over all admissible
partitions, we then get the conclusion. ∎

Then let’s consider the “lower bound” of field counting.

######  Lemma 5.11.

Let $\gamma\geq\gamma_{\underline{x}}$ be a fixed integer. For each
$\underline{x}\in V_{I}$, there exists a generating series
$\tau_{\gamma}(\underline{x},s)=\sum_{d}a_{d}d^{-s}$ such that
$\tau_{\gamma}(\underline{x},s)\leq\pi_{\gamma}(\underline{x},s)$ and such
that it defines a regular function in the open half-plane $\sigma>x_{1}^{-1}$
that admits the following analytic continuation. There exists regular
functions $f,g_{0},\dots,g_{\gamma-\delta_{\underline{x}}}$ in the closed
half-plane $\sigma\geq x_{1}^{-1}$ such that
$g_{\gamma-\delta_{\underline{x}}}$ can be chosen to be a nonzero constant and
$f(x_{0})\neq 0$, and such that

$\tau_{\gamma}(\underline{x},s)=f(s)(s-x_{0}^{-1})^{-\beta}\sum_{j=0}^{\gamma-\delta_{\underline{x}}}g_{j}(s)\big{(}\log\frac{1}{s-x_{\xi_{1}}^{-1}}\big{)}^{j}.$

###### Proof.

We give a construction of such a series as follows. Let
$\rho:\prod_{p}\mathbb{Z}_{p}^{*}\to G$ be a surjective homomorphism such that
$\delta_{\underline{x}}(\rho)=\delta_{\underline{x}}$, and such that
$\gamma(\rho)=\gamma$. The existence of such $\rho$ is given by Lemma 5.9. Let
$\gamma=n_{\xi_{1}}+\cdots+n_{\xi_{u}}$ be the partition corresponding to
$\rho$. Let $\rho_{0}$ be the restriction of $\rho$ to the places

$\\{p\mid\lvert G\rvert\\}\cup\\{p\in\mathcal{P}(\lvert
G\rvert)\mid\operatorname{im}(\rho|_{p})\cap\Omega=\emptyset\\}.$

Let $q$ be the largest rational prime such that $\rho_{0}|_{p}$ is nontrivial,
and let $d_{0}:=\vartheta_{\underline{x}}(\rho_{0})$. Define

$\tau_{\gamma}(\underline{x},s):=d_{0}^{-s}\sum_{\begin{subarray}{c}\underline{p}\in\mathcal{P}(1,\lvert
G\rvert)^{\gamma}\\\
q<p_{1}<\cdots<p_{\gamma}\end{subarray}}\prod_{k=1}^{u}\prod_{j=n_{\xi_{k-1}}+1}^{n_{\xi_{k}}}p_{j}^{-x_{\xi_{k}}s}\prod_{\begin{subarray}{c}p\in\mathcal{P}(\lvert
G\rvert)\\\ q<p\nmid p_{1}\cdots
p_{\gamma}\end{subarray}}(1+\sum_{\begin{subarray}{c}1\leq i\leq l\\\
i\neq\xi_{1},\dots,\xi_{u}\end{subarray}}\mathfrak{s}^{p}_{H_{i}}p^{-x_{i}}s),$

where $n_{\xi_{0}}:=0$ as a notation. For a fixed $d\in S$, if $a_{d}>0$, then
we can compute it by

$a_{d}=\sum_{d/d_{0}=e_{\xi_{1}}^{x_{\xi_{1}}}\cdots
e_{\xi_{u}}^{x_{\xi_{u}}}p_{1}^{x(F_{1})}\cdots
p_{n}^{x(F_{n})}}\prod_{j=1}^{n}\mathfrak{s}^{p_{i}}_{F_{n}}$

where $e_{\xi_{k}}$ is a square-free integer for each $k=1,\dots,u$ such that
$\omega(e_{\xi_{k}})=n_{\xi_{k}}$ and such that
$\gcd(e_{\xi_{k}},e_{\xi_{j}})=1$ for all $j,k=1,\dots,u$, and
$F_{1},\dots,F_{n}$ are all cyclic subgroups such that $F_{j}\neq H_{\xi_{k}}$
for all $j=1,\dots,n$ and $k=1,\dots,u$. Given such a decomposition
$d/d_{0}=e_{\xi_{1}}^{x_{\xi_{1}}}\cdots
e_{\xi_{u}}^{x_{\xi_{u}}}p_{1}^{x(F_{1})}\cdots p_{n}^{x(F_{n})}$, if the
product $\prod_{j=1}^{n}\mathfrak{s}^{p_{i}}_{F_{n}}$ is non-zero, we can
construct surjective homomorphisms $\rho:\prod_{p}\mathbb{Z}_{p}^{*}\to G$ as
follows. For each $p\mid e_{\xi_{k}}$, let $\rho_{p}$ be a fixed surjective
homomorphism $\mathbb{Z}_{q}^{*}\to H_{\xi_{k}}$. For each $1\leq j\leq n$,
let $\rho_{p_{j}}$ be a surjective homomorphism $\mathbb{Z}_{p_{j}}^{*}\to
F_{j}$. Note that
$\lvert\operatorname{Sur}(\mathbb{Z}_{p_{j}}^{*},F_{j})\rvert=\mathfrak{s}^{p_{j}}_{F_{j}}$.
Then let

$\rho^{\prime}:=\rho_{0}\prod_{j=1}^{n}\rho_{p_{j}}\prod_{k=1}^{u}\prod_{p\mid
e_{\xi_{k}}}\rho_{p}.$

First of all $\rho^{\prime}$ is surjective, because the set

$\bigcup_{p<q}\operatorname{im}(\rho^{\prime}\lvert_{p})\cup\bigcup_{k=1}^{u}\bigcup_{p\mid
e_{\xi_{k}}}\operatorname{im}(\rho^{\prime}_{p})$

is just the same as $\bigcup_{p\in\mathcal{P}}\operatorname{im}(\rho|_{p})$,
and it can generate the whole group according to our assumption. Second, for
each $1\leq j\leq n$, there are $\mathfrak{s}^{p_{j}}_{F_{j}}$ different local
surjective homomorphisms. So we can construct at least
$\mathfrak{s}^{p_{j}}_{F_{j}}$ different surjective homomorphisms
$\rho^{\prime}:\prod_{p}\mathbb{Z}_{p}^{*}\to G$ such that
$\vartheta_{\underline{x}}(\rho^{\prime})=d$. If
$d/d_{0}=e_{\xi_{1}}^{x_{\xi_{1}}}\cdots
e_{\xi_{u}}^{x_{\xi_{u}}}p_{1}^{x(F_{1})}\cdots p_{n}^{x(F_{n})}$ and
$d/d_{0}=f_{\xi_{1}}^{x_{\xi_{1}}}\cdots
f_{\xi_{u}}^{x_{\xi_{u}}}w_{1}^{x(T_{1})}\cdots w_{m}^{x(T_{m})}$ are two
different decompositions, then clearly either $\\{e_{\xi_{k}}\\}$ is not the
same as $\\{f_{\xi_{k}}\\}$, or $(p_{1},F_{1}),\dots,(p_{n},F_{n})$ and
$(w_{1},T_{1}),\dots,(w_{m},T_{m})$ are not the same, up to re-ordering. So we
see that

$a_{d}\leq\sum_{\begin{subarray}{c}\rho^{\prime}\in\operatorname{Sur}(\prod_{p}\mathbb{Z}_{p}^{*},G)\\\
\vartheta_{\underline{x}}(\rho^{\prime})=d\end{subarray}}\mathbf{1}_{(\Omega,\gamma)}(\rho^{\prime}).$

The proof of analytic properties is similar to that of
$\mu_{\gamma}(\underline{x},s)$. So we just leave it to the reader. ∎

######  Proposition 5.12.

Let $\gamma>\gamma_{\underline{x}}$ be a fixed positive integer. The
expression $\pi_{\gamma}(\underline{x},s)$ defines a continuous function in
$U_{I}$ that is a regular function in the open half-plane $\sigma>x_{1}$ for
each $\underline{x}\in V_{I}$.

  1. (1)

If $x_{1}<x_{\xi_{1}}$, then there exists a continuous function
$f:\bar{U}_{I}\to\mathbb{C}$ that is regular in the closed half-plane
$\sigma\geq x_{1}^{-1}$ such that $f(\underline{x},x_{1}^{-1})\neq 0$ and such
that

$\pi_{\gamma}(\underline{x},s)=f(\underline{x},s)(s-x_{1}^{-1})^{-\beta}.$

  2. (2)

Else if $x_{\xi_{1}}=x_{1}$, then there exists continuous functions
$f,f_{0},\dots,f_{\gamma-\delta_{\underline{x}}}:U_{I}\to\mathbb{C}$ that are
regular in the closed half-plane $\sigma\geq x_{1}^{-1}$ for each
$\underline{x}\in V_{I}$ such that
$f(x_{1}^{-1})f_{\gamma-\delta_{\underline{x}}}(x_{1}^{-1})\neq 0$ and such
that

$\pi_{\gamma}(\underline{x},s)=f(\underline{x},s)(s-x_{0}^{-1})^{-\beta}\sum_{j=0}^{\gamma-\delta_{\underline{x}}}f_{j}(\underline{x},s)\big{(}\log\frac{1}{s-x_{1}^{-1}}\big{)}^{j}.$

###### Proof.

Again, we start with the case when $G=C_{q}$, a cyclic group of prime order.
Since $\bar{G}=\\{\\{0\\},\Omega_{1}=G\backslash\\{0\\}\\}$, the only choice
of a nonempty $\Omega$ is $\Omega=\Omega_{1}$. In other words, for each
$K\in\mathcal{S}(G)$, we have $\mathbf{1}_{(\Omega,\gamma)}(K)=1$ if and only
if $K$ admits exactly $\gamma$ tamely ramified primes. Since

$\pi_{\gamma}(\underline{x},s)=\mu_{\gamma}(\underline{x},s)-1,$

we see that our conclusion holds by Lemma 5.7.

Now let $G$ be a finite abelian group. Assume that the conclusion holds for
each proper subgroup $H\subseteq G$. Again, let $\vartheta_{H,\underline{x}}$
be the induced invariant, i.e., for each $\rho:\prod_{p}\mathbb{Z}_{p}^{*}\to
H$, let $\vartheta_{H,\underline{x}}(\rho):=\vartheta_{\underline{x}}(\rho)$
according to the embedding $H\to G$. Also, for each proper subgroup $H$, let
$\Omega_{H}:=\Omega\cap H$. Then we see that

$\pi_{\gamma}(\underline{x},s)=\mu_{\gamma}(\underline{x},s)-\sum_{H\subsetneq
G}\pi_{H,\gamma}(\underline{x},s).$

This shows the existence of analytic continuation of
$\pi_{\gamma}(\underline{x},s)$. To be precise, there exists some integers
$\beta^{\prime},\gamma^{\prime}$ and some continuous functions
$f,g_{0},\dots,g_{\gamma^{\prime}}:\bar{U}_{I}\to\mathbb{C}$ that are regular
in the closed half-plane $\sigma\geq x_{1}^{-1}$ for each $\underline{x}\in
V_{I}$ such that

$\pi_{\gamma}(\underline{x},s)=f(\underline{x},s)(s-x_{0}^{-1})^{-\beta^{\prime}}\sum_{j=0}^{\gamma^{\prime}}g_{j}(s)\big{(}\log\frac{1}{s-x_{1}^{-1}}\big{)}^{j}.$

It suffices to compute $\beta^{\prime}$ and $\gamma^{\prime}$ and to give the
pole (limit) behaviour at $s=x_{1}$.

According to Lemma 5.10 and 5.11, we see that there exists
$\psi_{\gamma}(\underline{x},s)$ and $\tau_{\gamma}(\underline{x},s)$ such
that for each $\underline{x}\in V_{I}$, we have

$\tau_{\gamma}\leq\pi_{\gamma}(\underline{x},s)\leq\psi_{\gamma}(\underline{x},s).$

Moreover, $\psi$ and $\tau$ admits the following analytic continuation in the
closed half-plane $\sigma\geq x_{1}^{-1}$. For each $\underline{x}\in V_{I}$,
there exists some regular $g,g_{0},\dots,g_{\gamma-\delta_{0}}$ in the closed
half-plane $\sigma\geq x_{1}^{-1}$ such that $g_{\gamma-\delta_{0}}$ can be
chosen to be a positive constant and $g(\underline{x},x_{0}^{-1})\neq 0$, and
such that

$\tau_{\gamma}(\underline{x},s)=g(\underline{x},s)(s-x_{0}^{-1})^{-\beta}\sum_{j=0}^{\gamma-\delta_{\underline{x}}}g_{j}(\underline{x},s)(s-x_{\xi_{1}}^{-1})^{j}.$

On the other hand, there exists regular functions
$h,h_{0},\dots,h_{\gamma-\delta_{0}}$ in the closed half-plane $\sigma\geq
x_{1}^{-1}$ such that $h_{\gamma-\delta_{0}}$ can be chosen to be a positive
constant and $h(\underline{x},x_{0}^{-1})\neq 0$ , and such that

$\psi_{\gamma}(s)=h(\underline{x},s)(s-x_{0}^{-1})^{\beta}\sum_{j=0}^{\gamma-\delta_{0}}\big{(}\log\frac{1}{s-x_{i_{1}}^{-1}}\big{)}^{j}.$

Then Theorem 4.1 implies that the required analytic continuation. For example,
let’s say $x_{\xi_{1}}>x_{0}=x_{1}$. If $\beta^{\prime}>\beta$, then Theorem
4.1 would imply that

$N(\operatorname{Hom}(\prod_{p}\mathbb{Z}_{p}^{*},G),\vartheta_{\underline{x}};(\Omega,\gamma);x)=o(N(\operatorname{Sur}(\prod_{p}\mathbb{Z}_{p}^{*},G),\vartheta_{\underline{x}};(\Omega,\gamma);x)).$

The contradiction implies $\beta^{\prime}\geq\beta$. If
$\beta^{\prime}<\beta$, then by writing
$\pi_{\gamma}(\underline{x},s)=\sum_{d}a_{d}d^{-s}$ and
$\psi_{\gamma}(\underline{x},s)=\sum_{d}c_{d}d^{-s}$, we have

$\sum_{d<x}c_{d}=o(\sum_{d<x}a_{d})$

as $x\to\infty$. This contradicts Lemma 5.10, i.e., $a_{d}\leq c_{d}$ for all
$d\in S$. So $\beta^{\prime}\leq\beta$. And this proves that when
$x_{\xi_{1}}>x_{0}=x_{1}$, we have $\beta^{\prime}=\beta$, hence the analytic
continuation of $\pi_{\gamma}(\underline{x},s)$ in this case. The proof of
other cases are similar. And we are done. ∎

Finally, we use a statement to describe the general relation between the
ordering of number fields and the distribution of class groups in abelian
case.

######  Theorem 5.13.

Keep the notations $I,V_{I},\Omega$ as in Theorem 5.6. The Conjecture 1.2(2)
holds if and only if $x_{\xi_{1}}=x_{0}$ is the smallest among the parameters.

###### Proof.

First assume that $x_{\xi_{1}}=x_{0}$. When
$\gamma\geq\gamma_{\underline{x}}$, we see that

$R^{\gamma}_{\gamma+1}(\underline{x})=0.$

In other words, the statement of the Conjecture 1.2(1) holds for all
$\gamma\geq\gamma_{\underline{x}}$. If $\gamma<\gamma_{\underline{x}}$, then
by Lemma 5.7, Theorem 5.6 and 4.1, we see that

$\lim_{x\to\infty}\frac{N(\operatorname{Hom}(\prod_{p}\mathbb{Z}_{p}^{*},G),\vartheta_{\underline{x}};(\Omega,\gamma);x)}{N(\mathcal{S},\vartheta_{\underline{x}};x)}=\lim_{x\to\infty}\frac{c_{1}(\underline{x})x^{x_{1}^{-1}}(\log
x)^{\beta_{1}}(\log\log x)^{\gamma}}{c_{2}(\underline{x})x^{x_{1}^{-1}}(\log
x)^{\beta_{2}}},$

where $c_{1},c_{2}$ are continuous functions in $V_{I}$, and

$\displaystyle\beta_{1}=$
$\displaystyle\sum_{\begin{subarray}{c}i\neq\xi_{1},\dots,\xi_{u}\\\
x_{i}=x_{0}\end{subarray}}\beta(\Omega_{i}),$ $\displaystyle\beta_{2}=$
$\displaystyle\sum_{\begin{subarray}{c}1\leq i\leq l\\\
x_{i}=x_{0}\end{subarray}}\beta(\Omega_{i}).$

Since $x_{\xi_{1}}=x_{0}$ has the smallest value among the parameters, we see
that $\beta_{2}>\beta_{1}$. So

$\lim_{x\to\infty}\frac{c_{1}(\underline{x})x^{x_{1}^{-1}}(\log
x)^{\beta_{1}}(\log\log x)^{\gamma}}{c_{2}(\underline{x})x^{x_{1}^{-1}}(\log
x)^{\beta_{2}}}=0,$

and
$N(\mathcal{S},\vartheta_{\underline{x}};(\Omega,\gamma);x)=o(N(\mathcal{S},\vartheta_{\underline{x}};x))$,
as $x\to\infty$. Therefore, we’ve shown that the Conjecture 1.2(2) holds for
all $\gamma\geq 0$, and we are done. ∎

###  5.1. Example

Let $G=C_{2}^{n}$, where $n\geq 5$, and $\mathcal{S}:=\mathcal{S}(G)$ be the
set of $G$-fields. We want to consider the case when $\delta_{0}$ is non-
trivial defined in Definition 5.1. Write

$C_{2}^{3}=\\{g_{1},g_{2},\dots,g_{n}\mid
g_{1}^{2}=\cdots=g_{n}^{2}=1\\}^{\text{ab}},$

and let
$\Omega_{1}:=\\{g_{1}\\},\Omega_{2}=\\{g_{2}\\},\dots,\Omega_{n}:=\\{g_{n}\\},\dots$.
According to Definition 3.2, we let $\vartheta_{\underline{x}}$, where
$\underline{x}\in\mathbb{R}_{+}^{2^{n}-1}$, be the invariant of fields
$K\in\mathcal{S}$ such that $x_{i}$ corresponds to $\Omega_{i}$. In other
words, let $p$ be an odd prime and let $\rho_{p}:\mathbb{Z}_{p}^{*}\to G$ be a
homomorphism. If $\operatorname{im}(\rho_{p})=\langle\Omega_{i}\rangle$, then
$\vartheta_{\underline{x}}(\rho_{p})=p^{x_{i}}$. Let $\Omega=G-\\{1,g_{1}\\}$.
Let $V=\\{\underline{x}\in\mathbb{R}_{+}^{2^{n}-1}\mid x_{1}\leq
x_{2}\leq\cdots\leq x_{2^{n}-1}\\}$ and let $I:=\\{2\\}$. Then
$V_{I}=\\{\underline{x}\in V\mid x_{1}=x_{2}<x_{3}=\cdots=x_{2^{n}-1}\\}$.
Note that if a homomorphism $\rho:\prod_{p}\mathbb{Z}_{p}^{*}\to G$ is a
restriction of some Artin reciprocity map
$\tilde{\rho}:\mathbf{C}_{\mathbb{Q}}\to C_{4}$, which corresponds to a
$C_{4}$-field in $\mathcal{S}$, then $\rho$ has to be surjective.

Let $p$ be an odd prime and let $\rho_{p}:\mathbb{Z}_{p}^{*}\to G$ be a
homomorphism. We see that $\vartheta_{t}(\rho_{p})=p$ if and only if
$\operatorname{im}(\rho_{p})=\langle g_{1}\rangle$ or $\langle g_{2}\rangle$.
On the other hand, since any homomorphism $\rho_{2}:\mathbb{Z}_{2}^{*}\to G$
factors through $C_{2}\times C_{2}$, we see that $\operatorname{im}(\rho_{2})$
has rank at most $2$ as an abelian $2$-group.

Let’s construct a homomorphism $\rho:\prod_{p}\mathbb{Z}_{p}^{*}\to G$ as
follows. Let $p_{1},\dots,p_{n-2}$ be $n-2$ odd primes. For each
$i=1,\dots,n-2$, let $\rho_{p_{i}}:\mathbb{Z}_{p_{i}}^{*}\to G$ be a
homomorphism such that $\operatorname{im}(\rho_{p_{i}})=\langle g_{i}\rangle$.
Let $\rho_{2}:\mathbb{Z}_{2}^{*}\to G$ be a homomorphism such that
$\operatorname{im}(\rho_{2})=\langle g_{n-1},g_{n}\rangle$. It is clear that
$\rho:=\rho_{2}\cdot\prod_{i=1}^{n-2}\rho_{p_{i}}$ is a surjective
homomorphism onto $G$. Moreover, $\delta_{t}(\rho)=n-4$ because $\rho$ is
surjective itself and there are exactly $n-4$ odd primes $p_{3},\dots,p_{n-2}$
satisfying the condition $\vartheta_{t}(\rho|_{p})=p^{t}$.

Let’s show that $\delta_{t}\geq n-4$. Assume for contradiction that
$\delta_{t}<n-4$. Then there exists some surjective homomorphism
$\rho:\prod_{p}\mathbb{Z}_{p}^{*}\to G$ such that $\delta_{t}(\rho)<n-4$. Let
$p_{1},\dots,p_{m}$ be all the odd primes such that
$\vartheta_{t}(\rho_{p})>1$. Since $\delta_{t}(\rho)<n-4$, there are at most
$n-5$ primes, say $p_{1},\dots,p_{n-5}$, satisfying the condition
$\vartheta_{t}(\rho_{p})=p^{t}$. But this means that
$\vartheta_{t}(\rho_{p_{i}})=p_{i}$ for all $i>n-5$, in other words,
$\operatorname{im}(\rho_{p_{i}})\subseteq\langle g_{1},g_{2}\rangle$. By
comparing the rank as an abelian $2$-group, we see that

$\langle
g_{1},g_{2}\rangle\cup\operatorname{im}(\rho_{2})\cup\bigcup_{i=1}^{n-5}\operatorname{im}(\rho_{p_{i}})$

cannot generate $G$. This shows that $\delta_{t}\geq n-4$, hence
$\delta_{t}=n-4$. Moreover since $\gamma(\rho)=n-3$, we see that
$\gamma_{t}\leq n-3$. According to Theorem 5.6 and Theorem 4.1, for
$\gamma\geq n-3$, we have

$N(\mathcal{S},\vartheta_{t};(\Omega,\gamma);x)\sim c(t)x(\log\log
x)^{\gamma-\delta_{t}},$

where $c(t)>0$ is a continuous function.

## References

  * [1] Manjul Bhargava, Arul Shankar, and Jacob Tsimerman.  On the davenport–heilbronn theorems and second order terms.  Inventiones mathematicae, 193(2):439–499, 2013.
  * [2] Hubert Delange.  Généralisation du théorème de Ikehara.  Annales scientifiques de l’École Normale Supérieure, 3e série, 71(3):213–242, 1954.
  * [3] Ellen Eischen, Ling Long, Rachel Pries, and Katherine Stange.  Directions in Number Theory: Proceedings of the 2014 WIN3 Workshop, volume 3.  Springer, Cham, 01 2016.
  * [4] Henryk Iwaniec and Emmanuel Kowalski.  Analytic Number Theory.  American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, Rhode Island, 2004.
  * [5] H.L. Montgomery and R.C. Vaughan.  Multiplicative Number Theory I: Classical Theory.  Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2006.
  * [6] W. Narkiewicz.  Elementary and Analytic Theory of Algebraic Numbers.  Springer, Berlin Heidelberg, 2014.
  * [7] Weitong Wang.  Distribution of the bad part of the class group.  2022\.
  * [8] Melanie Matchett Wood.  On the probabilities of local behaviors in abelian field extensions.  Compositio Mathematica, 146(1):102–128, 2010.