Example 2.10.
When \(\chi=\mathds{1}\text{,}\) then, for \(\Re(s)\gt 1\)
\begin{gather*}
L(s,\mathds{1})=\sum_{n=1}^\infty\frac{\mathds{1}(n)}{n^s}=\sum_{n=1}^\infty\frac{1}{n^s}=\zeta(s).
\end{gather*}
Section 2.2 \(L\)–functions
After Riemann and Dirichlet’s groundbreaking works, Number Theorists often consider functions defined in the form of a formal Dirichlet Series
\begin{gather}
\sum_{n=1}^\infty\frac{a_n}{n^s}.\tag{2.1}
\end{gather}
The coefficients \(a_n\) are usually \(a_n=f(n)\text{,}\) where \(f:\N\longrightarrow\C\) is an arithmetic function, and \(s\) has a sufficiently large real part. For instance, the simplest case is when \(f(n)=1\) for all \(n\in\N\text{.}\) Then \(\sum_{n=1}^\infty a_nn^{-s}=\zeta(s)\) is the Riemann Zeta function, defined for \(\Re(s)\gt 1\text{.}\) Dirichlet considered a more general situation when \(a_n=\chi(n)\) for a Dirichlet character \(\chi\text{.}\) He used the notation \(L(s,\chi)\colonequals\sum_{n=1}^\infty\chi(n)n^{-s}\text{,}\) now widely known as the Dirichlet \(L\)–functions. In the next two sections, we will discuss the Riemann Zeta and the Dirichlet \(L\)–functions in more detail.
Although Mathematicians never fixed a definition of an \(L\)–function, we expect specific properties to be satisfied. Below are the properties one expects an \(L\)–function should hold: these are modeled on well-known and widely used examples of \(L\)–functions like the Riemann Zeta function.
-
An \(L\)–function is usually a function \(f(s)\) of a complex variable \(s\text{,}\) defined and absolutely convergent for all \(s\in\C\) with sufficiently large real part.
-
An \(L\)–function \(f(s)\) should admit a meromorphic continuation to all of \(\C\text{.}\)
-
An \(L\)–function should satisfy a functional equation and be expressible as a Dirichlet series attached to a sequence of arithmetic importance.
-
An \(L\)–function should admit an Euler product expression of the form\begin{gather*} f(s)=\prod_pf_p(s), \end{gather*}where \(f_p(s)\) is called the local Euler factor at a prime \(p\text{.}\)
Subsection 2.2.1 The Riemann Zeta function
The Riemann Zeta function, denoted as \(\zeta(s)\text{,}\) is a function of one complex variable \(s\text{,}\) defined as following Dirichlet series
\begin{gather}
\zeta(s)\colonequals\sum_{n=1}^{\infty}\frac{1}{n^s}\tag{2.2}
\end{gather}
Fact 1. The Riemann Zeta function \(\zeta(s)\) admits a meromorphic continuation to all of \(\C\) with just one simple pole at \(s=1\) with residue \(1\text{.}\)
Fact 2. Following is the functional equation of \(\zeta(s)\)
\begin{gather}
\zeta(s) = 2^s \pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s)\tag{2.3}
\end{gather}
where \(\Gamma(z)\) is the Gamma function defined as
\begin{gather*}
\Gamma(z)=\int_0^\infty t^{z-1}e^{-t}\rmd t\quad\Re(z)\gt 0.
\end{gather*}
Fact 3. The Euler product of \(\zeta(s)\) is as follows:
\begin{gather}
\zeta(s)=\prod_p\bparen{1-\frac{1}{p^s}}^{-1}.\tag{2.4}
\end{gather}
Fact 4. For a negative odd integer \(1-2n\text{,}\) we have
\begin{gather}
\zeta(1-2n)=-\frac{B_{2n}}{2n}\tag{2.5}
\end{gather}
where \(B_n\) is the \(n^{\rm th}\) Bernoulli number defined as follows:
\begin{gather*}
\frac{t}{e^t-1}=\sum_{n=0}^\infty\frac{B_n}{n!}t^n.
\end{gather*}
So, putting \(s=2n\) in the functional equation (2.3), we get
\begin{gather}
\zeta(2n)=(-1)^{n+1}\frac{(2\pi)^{2n}B_{2n}}{2(2n!)}.\tag{2.6}
\end{gather}
In particular, for \(n=1\text{,}\) we have \(\zeta(2)=\pi^2/6\text{.}\)
Subsection 2.2.2 \(L\)–functions attached to Dirichlet characters
A Dirichlet \(L\)–function is defined as the Dirichlet series attached to the sequence \(\{\chi(n)\}_{n\geq1}\) of values of a Dirichlet character. Usually denoted by \(L(s,\chi)\text{,}\) it is defined as follows:
\begin{gather*}
L(s,\chi)\colonequals\sum_{n=1}^\infty\frac{\chi(n)}{n^s}\qquad\Re(s)\gt 1.
\end{gather*}
\(L(s,\chi)\) is absolutely convergent and holomorphic on \(\Re(s)\gt 1\) since \(\abs{\chi(n)}\in\{0,1\}\) for all \(n\in\N\text{.}\) We give some examples below.
Let \(v_p\) be the normalized \(p\)–adic valuation on \(\Z\text{,}\) i.e., \(v_p(n)\) is the maximum exponent of \(p\) in the prime factorization of \(n\text{.}\) Then,
\begin{align}
L(s,\chi)\amp=\sum_{n=1}^\infty\frac{\chi(n)}{n^s}=\sum_{n=1}^\infty\prod_{p}(\chi(p)p^{-s})^{v_p(n)}\notag\\
\amp=\prod_p\sum_{e=0}^\infty (\chi(p)p^{-s})^e=\prod_p\bparen{1-\frac{\chi(p)}{p^s}}^{-1}.\tag{2.7}
\end{align}
Example 2.11.
For \(\chi=\mathds{1}_N\) for some \(N\gt 1\text{,}\) we have
\begin{gather*}
L(s,\mathds{1}_N)=\sum_{\gcd(n,N)=1}\frac{1}{n^s}=\prod_{p\,\nmid N}\sum_{e=0}^\infty p^{-es}=\zeta(s)\prod_{p\mid N}\bparen{1-\frac{1}{p}}.
\end{gather*}
Therefore, \(L(s,\mathds{1}_N)\) has a simple pole coming from the factor with residue
\begin{gather*}
\Res_{s=1}L(s,\mathds{1})=\lim_{s\to1^+}(s-1)\zeta(s)\prod_{p\mid N}\bparen{1-\frac{1}{p}}=\prod_{p}\bparen{1-\frac{1}{p}}=\frac{\varphi(N)}{N}.
\end{gather*}
When \(\chi\) is not principal, \(L(s,\chi)\) analytically continues to all of \(\C\) [31, Theorem 10.7] and \(L(1,\chi)\neq0\) [47, Corollary 4.4]. This nonvanishing result is crucial in proving the infinitude of primes in arithmetic progressions. See [1, Chapter 7] for more details. The functional equation satisfied by a Dirichlet \(L\)–function is complicated. Let \(\chi\) be a primitive Dirichlet character modulo \(f_\chi\text{,}\) the conductor of \(\chi\text{.}\) Define the quantities
\begin{gather*}
\tau(\chi)\colonequals\sum_{n=1}^{f_\chi}\chi(n)e^{2\pi i n/f_\chi}\qquad\varepsilon(\chi)\colonequals\frac{\tau(\chi)}{i^a\sqrt{f_\chi}}
\end{gather*}
where \(a=(1-\chi(-1))/2\text{.}\) The sum \(\tau(\chi)\) is known as the Gauss sum associated to \(\chi\text{.}\) Then,
\begin{gather}
L(s,\chi) = \varepsilon(\chi) 2^s \pi^{s-1} q^{1/2-s} \sin \left( \frac{\pi}{2} (s + a) \right) \Gamma(1-s) L(1-s, \overline{\chi}),\tag{2.8}
\end{gather}
where \(\overline{\chi}\) is the complex conjugate character given my \(n\mapsto\overline{\chi(n)}\text{.}\) For a proof of the functional equation (2.8), we refer to [31, p. 333].
There are special value formulae for Dirichlet \(L\)–functions analogous to the formulae (2.5) and (2.6). For a primitive Dirichlet character \(\chi\) modulo \(f_\chi\text{,}\) the generalized Bernoulli numbers \(B_{n,\chi}\) are defined as follows:
\begin{gather}
\sum_{a=1}^{f_\chi}\frac{\chi(a)te^{at}}{e^{f_\chi t}-1}=\sum_{n=0}^\infty \frac{B_{n,\chi}}{n!}t^n.\tag{2.9}
\end{gather}
Theorem D ([47] Theorem 4.2).
For \(n\geq1\text{,}\) we have
\begin{gather}
L(1-n,\chi)=-\frac{B_{n,\chi}}{n}.\tag{2.10}
\end{gather}
These special value formulae will be used in Chapter 4.
Subsection 2.2.3 Dedekind Zeta function attached to a number field
A number field \(K\) is a finite extension of \(\Q\) (usually written as \(K/\Q\)). The dimension of \(K\) as a \(\Q\)-vector space is called the degree of the number field and is denoted by \([K:\Q]\text{.}\) For instance, \(\Q(i)\text{,}\) the smallest field containing the imaginary unit \(i\text{,}\) is a number field of degree 2 over \(\Q\text{.}\) The ring of integers in \(K\) is a Dedekind domain and, is denoted by \(\mathcal{O}_K\text{.}\) The absolute norm of an ideal \(\mf{a}\in\mathcal{O}_K\) is defined to be \(N(\mf{a})\colonequals[\mathcal{O}_K:\mf{a}]\text{.}\) We assume the basics of Algebraic Number Theory and some Commutative Algebra. For details, the readers may refer to standard texts like [47, 29, 34, 3].
The Dedekind Zeta function, first introduced by Richard Dedekind in his supplement [11] to Dirichlet’s Vorlesungen über Zahlentheorie, is defined as
\begin{gather}
\zeta_K(s)\colonequals\sum_{\mf{a}}\frac{1}{N(\mf{a})^s}=\prod_{\mf{p}}\bparen{1-\frac{1}{N(\mf{p})^s}}^{-1}\qquad\Re(s)\gt 1.\tag{2.11}
\end{gather}
The second equality follows from the unique factorization of ideals in \(\mathcal{O}_K\text{.}\) If the ideal \(p\mathcal{O}_K\) factorizes into primes as \(p\mathcal{O}_K=\mf{p}_1^{e_1}\mf{p}_2^{e_2}\cdots\mf{p}_g^{e_g}\text{,}\) then \(\sum_{i=1}^ge_if_i=[K:\Q]\equalscolon n\) [29, Theorem 21], where \(p^{f_i}=\#\mathcal{O}_K/\mf{p}_i\text{.}\) In particular, for a Galois number field \(K/\Q\text{,}\) \(e_1=e_2=\cdots=e_g\equalscolon e\text{,}\) \(f_1=f_2=\cdots=f_g\equalscolon f\text{,}\) and \(efg=n\) [29, Corollary to Theorem 23, p. 50]. Therefore, at most \(n\) primes are lying above an integer prime \(p\text{.}\) Since \(\mathcal{O}_K\) is a Dedekind domain, \(\mathcal{O}_K/\mf{p}\) is a finite field of characteristic \(p\) and hence \(N(\mf{p})=[\mathcal{O}_K:\mf{p}]\geq p\text{.}\) Combining all these, we get
\begin{gather*}
\sum_{\mf{p}}\abs{\log\bparen{1-\frac{1}{\mf{p}^s}}}\leq n\sum_{p}\abs{\log\bparen{1-\frac{1}{p^s}}}.
\end{gather*}
Consequently, the absolute convergence of the Euler product expression of the Riemann Zeta (equation (2.4) in §2.1) for \(\Re(s)\gt 1\) implies the absolute convergence for the Euler product expression of the Dedekind Zeta function.
Remark 2.12.
The Dedekind Zeta function is a generalization of the Riemann Zeta function. For \(K=\Q\text{,}\) the ring of integers is \(\Z\text{;}\) hence, the ideals are just \(\ideal{n}=n\Z\) for positive integers \(n\text{.}\) Therefore, \(N(\ideal{n})=[\Z:\ideal{n}]=\#\Z/\ideal{n}=n\text{.}\) Thus,
\begin{gather*}
\zeta_\Q(s)=\sum_{\ideal{n}}\frac{1}{N(\ideal{n})^s}=\sum_{n=1}^\infty\frac{1}{n^s}=\zeta(s).
\end{gather*}
Example 2.13.
We give a nontrivial example. Consider the number field \(K=\Q(i)\text{.}\) The primes of the ring of Gaussian integers \(\Z[i]\) are well understood (see [34, Chapter I, Theorem 1.4]): the prime \(2\) is totally ramified, \(2\mathcal{O}_K=\ideal{1-i}^2\) and hence \(N(\ideal{1-i})=2\text{.}\) If \(p\) is \(1\pmod{4}\text{,}\) \(p\mathcal{O}_K=\mf{p}_1\mf{p}_2\) for two different primes \(\mf{p}_i\text{,}\) and \(N(\mf{p}_i)=p\) for \(i=1,2\text{.}\) For \(p\equiv3\pmod{4}\text{,}\) \(p\) is inert and hence \(p\mathcal{O}_K=\mf{p}\) for some prime ideal \(\mf{p}\) in \(\mathcal{O}_K\) such that \(N(\mf{p})=p^2\text{.}\) Therefore, by the Euler product identity of \(\zeta_{\Q(i)}(s)\text{,}\) as in (2.11), we have
\begin{align}
\zeta_{\Q(i)}(s)\amp=\prod_{\mf{p}}\bparen{1-\frac{1}{N(\mf{p})^s}}^{-1}\notag\\
\amp=\bparen{1-\frac{1}{2^s}}^{-1}\prod_{\substack{p\equiv1\pmod{4}\\\ideal{p}=\mf{p}_1\mf{p_2}}}\bparen{1-\frac{1}{p^s}}^{-2}\prod_{\substack{p\equiv3\pmod{4}\\\ideal{p}=\mf{p}}}\bparen{1-\frac{1}{p^{2s}}}^{-1}\notag\\
\amp=\prod_{p}\bparen{1-\frac{1}{p^s}}^{-1}\prod_{p\equiv1\pmod{4}}\bparen{1-\frac{1}{p^s}}^{-1}\prod_{p\equiv3\pmod{4}}\bparen{1+\frac{1}{p^s}}^{-1}.\tag{2.12}
\end{align}
Where the second step follows after utilizing the identity
\begin{gather*}
(1-p^{-2s})=(1+p^{-s})(1-p^{-s})
\end{gather*}
and regrouping the terms. Note that in the product (2.12),
\begin{gather*}
\prod_{p\equiv1\pmod{4}}\bparen{1-\frac{1}{p^s}}^{-1}\prod_{p\equiv3\pmod{4}}\bparen{1+\frac{1}{p^s}}^{-1}=\sum_{n=1}^{\infty}\frac{(-1)^{(n-1)/2}}{n^s}=L(s,\chi_{-4}),
\end{gather*}
where \(\chi_{-4}\) denotes the nontrivial character modulo \(4\) defined by \(\chi_{-4}(1)=1\) and \(\chi_{-4}(3)=-1\text{.}\) Therefore, \(\zeta_{\Q(i)}(s)=\zeta(s)L(s,\chi)\text{.}\) This factorization is quite interesting and will play a crucial role in this thesis. The immediate consequence is that \(\zeta_{\Q(i)}(s)/\zeta(s)\) is an entire function (by [31, Theorem 10.7]). So, one might be interested in classifying number fields \(K\) such that \(\zeta_K/\zeta(s)\) is entire. This is a particular case of a weaker form of Artin Conjecture, often called the Dedekind Conjecture, which predicts that for an extension of number fields \(L/K\text{,}\) the quotient \(\zeta_L(s)/\zeta_K(s)\) is entire. This is true for Galois extensions, known as the Aramata-Brauer Theorem [32, Chapter 2, Theorem 3.1]. For a non-Galois version of the Dedekind Conjecture, see [32, Chapter 2, Theorem 4.1].
Subsection 2.2.4 Cyclotomic fields
The \(n^{\rm th}\) cyclotomic field is defined by adjoining the complex \(n^{\rm th}\) roots of unity to \(\Q\text{.}\) Let \(\mu_n\) denote the set of all \(n^{\rm th}\) complex roots of unity. Then we write \(\Q(\mu_n)\) to denote the \(n^{\rm th}\) cyclotomic field.
We start with an observation about cyclotomic fields that will motivate the necessity of this section to prove our main result. Note that there is something special about \(\Q(i)\text{.}\) We can write \(\Q(i)=\Q(\mu_4)\text{,}\) i.e., the 4th cyclotomic field. Also, \(\mathcal{X}(4)=\{\mathds{1},\chi_{-4}\}\text{.}\) Therefore,
\begin{gather}
\zeta_{\Q(\mu_4)}(s)=\prod_{\chi\in\mathcal{X}(4)}L(s,\chi).\tag{2.13}
\end{gather}
It turns out that the factorization in (2.13) is true in general:
Proposition 2.14. see Theorem 2.16.
Let \(n\) be a positive integer and \(\Q(\mu_n)\) be the \(n^{\rm th}\) cyclotomic field. Then
\begin{gather}
\zeta_{\Q(\mu_n)}(s)=\prod_{\chi\in\mathcal{X}(n)}L(s,\chi).\tag{2.14}
\end{gather}
In this section, we will sketch a proof of a slightly general version of Proposition 2.14, which will be used to prove our main result. Before that, we state some basic facts about cyclotomic fields without proof. For details and proofs, refer to [47, 29, 34].
Fact 1. The \(n^{\rm th}\) cyclotomic field is a number field of degree \(\varphi(n)\text{,}\) where \(\varphi(n)\) is the Euler totient function, as mentioned before (see [29, Corollary 1, p. 13]).
Fact 2. The ring of algebraic integers, i.e., the set of elements in \(\Q(\mu_n)\) that is a zero of a monic polynomial with integer coefficients, is \(\Z[\mu_n]\) (see [47, Theorem 2.6]).
Fact 3. When \(n=p^\nu\) is a prime power, \(\ideal{1-\zeta_{p^\nu}}\text{,}\) where \(\zeta_{p^\nu}\) is a primitive \(p^\nu\)-th root of unity, is a prime ideal, and \(\ideal{1-\zeta_p}^{\varphi(p^\nu)}=\ideal{p}\text{.}\) As a result, \(p\) is totally ramified in \(\Q(\mu_{p^\nu})\) (see [34, Chapter I, Lemma 10.1]).
Fact 4. For \(n\gt 1\text{,}\) we write \(\Q(\mu_n)^+\) for \(\Q(\zeta_n+\zeta_n^{-1})\text{,}\) where \(\zeta_n\) is a primitive \(n^{\rm th}\) root of unity. The extension \(\Q(\mu_n)^+/\Q\) is of degree \(\varphi(n)/2\) and all its embeddings in \(\C\) takes values in \(\R\text{.}\) Such number fields are called totally real. The \(n^{\rm th}\) cyclotomic field \(\Q(\mu_n)\) is a degree 2 extension over \(\Q(\mu_n)^+\text{.}\) Number fields that are degree 2 extensions of a totally real field are called a CM extension. Therefore, \(\Q(\mu_n)\) is CM over \(\Q(\mu_n)^+\text{.}\)
Being the splitting field of the polynomial \(x^n-1\in\Z[x]\text{,}\) \(\Q(\mu_n)\) is a Galois extension of \(\Q\) and \(\Gal(\Q(\mu_n)/\Q)\cong(\Z/n\Z)^\times\text{,}\) where the isomorphism sends \(\sigma\in\Gal(\Q(\mu_n)/\Q)\) to \(a\pmod{n}\in(\Z/n\Z)^\times\) such that \(\sigma(\zeta_n)=\zeta_n^a\) for the primitive \(n^{\rm th}\) of unity \(\zeta_n=e^{2\pi i/n}\text{.}\) Therefore, we may identify the group of characters of \(\Gal(\Q(\mu_n)/\Q)\) with the group \(\mathcal{X}(n)\) (as defined in Lemma 2.7) of primitive Dirichlet characters whose conductors divide \(n\text{.}\)
The following Lemma is crucial to set the stage for the main result of this chapter: Recall that in Lemma 2.7, we defined \(\mathcal{X}(n)\) to be the group of primitive Dirichlet characters whose conductors divide \(n\text{.}\)
Lemma 2.15.
For any subgroup \(H\subseteq \mathcal{X}(n)\text{,}\) define:
\begin{gather}
H^\perp\colonequals\{\sigma\in\Gal(\Q(\mu_n)/\Q)\colon \chi(\sigma)=1,\chi\in H\}.\tag{2.15}
\end{gather}
Conversely, for any subfield \(K\) of \(\Q(\mu_n)\text{,}\) define:
\begin{gather}
H_K\colonequals\{\chi\in \mathcal{X}(n)\colon \chi(\sigma)=1,\sigma\in\Gal(\Q(\mu_n)/K)\}.\tag{2.16}
\end{gather}
There is a one-one correspondence between subgroups of \(\mathcal{X}(n)\) and subfields \(K\subset\Q(\mu_n)\) given by
\begin{align}
H \amp \mapsto \Q(\mu_n)^{H^\perp}\notag\\
H_K\amp\mapsfrom K.\tag{2.17}
\end{align}
Proof.
This is a special case of a more general result [47, Proposition 3.3], which is an analogue of the Fundamental Theorem of Galois Theory [28, Chapter VI, Theorem 1.1] in the context of characters of finite abelian groups. Let \(K=\Q(\mu_n)^{H^\perp}\text{.}\) By the Fundamental Theorem of Galois Theory, \(\Gal(\Q(\mu_n)/K)=H^{\perp}\text{.}\) Identifying \(\widehat{\Gal(\Q(\mu_n)/\Q)}\) with \(\widehat{(\Z/n\Z)^{\times}}\cong\mathcal{X}(n)\text{,}\) we may think of \(H\) as a subgroup of \(\widehat{\Gal(\Q(\mu_n)/\Q)}\text{.}\) By [47, Propositions 3.3 and 3.4] and by Poincaré Duality [47, Corollary 3.2], \(H_K=(H^\perp)^\perp=H\text{.}\) This implies that the correspondences (2.17) are indeed inverses of each other and hence finishes the proof.
The subgroup \(H\) can be regarded as the character group of the Galois group of the fixed field \(K\colonequals\Q(\mu_n)^{H^\perp}\) over \(\Q\text{.}\) We are ready to state a general version of Proposition 2.14. The proof is an elaborated version of the proof of [47, Theorem 4.3].
Theorem 2.16.
Let \(H\) be a subgroup of \(\mathcal{X}(n)\) and \(K=\Q(\mu_n)^{H^\perp}\) be the corresponding fixed field. Then
\begin{gather}
\zeta_K(s)=\prod_{\chi\in H}L(s,\chi).\tag{2.18}
\end{gather}
We need the following lemma for the proof: Let \(v_p(n)\) denote the maximum exponent \(e\) such that \(p^e\mid n\) (see Definition 3.1 to see how \(v_p\) extends to all of \(\Q\setminus\{0\}\text{.}\))
Lemma 2.17.
Let \(n\) be a positive integer and \(n'\colonequals n/p^{v_p(n)}\text{.}\) Then \(\Q(\mu_{n'})\) is the maximal subfield of \(\Q(\mu_n)\) which is unramified over \(\Q\) at \(p\text{.}\)
Proof.
We know that \(p\) is ramified in \(\Q(\mu_n)\) if and only if \(p\mid n\) [47, Propostition 2.3]. Since \(p\nmid n'\text{,}\) \(\Q(\mu_{n'})/\Q\) is unramified at \(p\text{.}\) Now, \(\Q(\mu_n)\) is the compositum of \(\Q(\mu_{n'})\) and \(\Q(p^{v_p(n)})\) and \(\Q(\mu_{n'})\cap\Q(p^{v_p(n)})=\Q\text{,}\) since \(\gcd(n',p^{v_p(n)})=1\) [47, Proposition 2.4]. Therefore, any nontrivial subextension \(K\text{,}\) such that \(\Q(\mu_n)/K/\Q(\mu_{n'})\text{,}\) contains a subfield of \(\Q(p^{v_p(n)})\) which properly contains \(\Q\text{.}\) Any nontrivial subfield of \(\Q(p^{v_p(n)})\) is ramified at \(p\text{,}\) which forces \(K=\Q(\mu_{n'})\) and finishes the proof.
Proof of Theorem 2.16.
Using the Euler product (2.11) of Dedekind zeta functions, we can write \(\zeta_K(s)\) as a product over rational primes as follows
\begin{gather}
\zeta_K(s)=\prod_{\mf{p}}\bparen{1-\frac{1}{N(\mf{p})^s}}^{-1}=\prod_{p}\prod_{\mf{p}|p}\bparen{1-\frac{1}{N(\mf{p})^s}}^{-1}.\tag{2.19}
\end{gather}
In the second product, the inner product runs over all prime ideals \(\mf{p}\) over \(p\text{,}\) and we denote that using the standard symbol \(\mf{p}|p\text{.}\) On the other hand, the left-hand side of equation (2.18) can be written as a product over integer primes as well, using the Euler product (2.7) of Dirichlet \(L\)–functions:
\begin{gather}
\prod_{\chi\in H}L(s,\chi)=\prod_{p}\prod_{\chi\in H}\bparen{1-\frac{\chi(p)}{p^s}}^{-1}.\tag{2.20}
\end{gather}
\begin{gather}
\prod_{\mf{p}|p}\bparen{1-\frac{1}{N(\mf{p})^s}}^{-1}= \prod_{\chi\in H}\bparen{1-\frac{\chi(p)}{p^s}}^{-1}.\tag{2.21}
\end{gather}
for each rational prime \(p\text{.}\) Note that \(\Gal(\Q(\mu_n)/\Q)\) is abelian; hence, any subgroup is normal. Hence, by the Fundamental Theorem of Galois Theory [28, Chapter VI, Theorem 1.1], \(K/\Q\) is a Galois extension.
Fix an integer prime \(p\) and Let \(e_p\) and \(f_p\) denote the ramification index and inertia degree respectively and \(g_p\) denote the number primes in \(\mathcal{O}_K\) lying over \(p\text{.}\) Then,
\begin{align}
\prod_{\mf{p}|p}\bparen{1-\frac{1}{N(\mf{p})^s}}^{-1} \amp =\bparen{1-\frac{1}{p^{f_ps}}}^{-g_p}\tag{2.22}
\end{align}
for each integer prime \(p\text{.}\) First, suppose that \(p\nmid n\text{.}\) Then \(\Q(\mu_n)\) and, in particular, \(K\) is unramified over \(\Q\) at \(p\text{.}\) Consequently, \(e_p=1\) and hence \([K:\Q]=f_pg_p\text{.}\) Since \(K/\Q\) is unramified at \(p\) and \(\Gal(K/\Q)\) is a subgroup of \((\Z/n\Z)^\times\text{;}\) hence abelian, the Frobenius map \(x\mapsto x^p\) in \(\Gal(\mathcal{O}_K/\mf{p}/\F_p)\text{,}\) which is of order \(f_p\text{,}\) lifts to an order \(f_p\) element, denoted as \(\Frob{p}\text{,}\) in \(\Gal(K/\Q)\text{.}\) Regard \(H\) as the group of characters of \(\Gal(K/\Q)\text{.}\) For any \(\chi\in H\text{,}\) note that \(\chi(\Frob{p})\) is a complex \(f_p^{\rm th}\) root of unity. Consider the map \(\Phi:H\longrightarrow \mu_{f_p}\) given by \(\chi\mapsto\chi(\Frob{p})\text{,}\) where we think of \(\mu_{f_p}\) as a multiplicative subgroup of \(\C^\times\text{.}\) Suppose that \(\chi\in\ker(\Phi)\text{,}\) i.e., \(\chi(\Frob{p})=1\text{.}\) Then \(\chi\) can be considered as a character on \(\Gal(K/\Q)/\ideal{\Frob{p}}\text{,}\) where \(\ideal{\Frob{p}}\) is the cyclic group of order \(f_p\) generated by \(\Frob{p}\text{.}\) Conversely, any character \(\chi\) of \(\Gal(K/\Q)/\ideal{\Frob{p}}\) can be extended to a character \(\chi\in H\) such that \(\chi(\Frob{p})=1\text{.}\) Therefore, \(\#\ker(\Phi)=\#\Gal(K/\Q)/f_p=g_p\text{.}\) By the First Isomorphism Theorem of groups [28, Chapter I, p. §3], \(\Im(\Phi)\subset\mu_{f_p}\) has \(f_p=\#\mu_{f_p}\) elements; hence \(\Im(\Phi)=\mu_{f_p}\text{.}\) This shows that \(\Phi\) is surjective and as \(\chi\) varies over \(H\text{,}\) \(\chi(p)=\chi(\Frob{p})\) varies over all the complex \(f_p^{\rm th}\) roots of unity, where each of them occur exactly \(g_p\) times (via the isomorphism \((\Z/n\Z)^\times\cong\Gal(\Q(\mu_n)/\Q)\text{,}\) \(p\pmod{n}\in(\Z/n\Z)^\times\) corresponds to \(\Frob{p}\in\Gal(K/\Q)\)). Therefore,
\begin{gather}
\prod_{\chi\in H}\bparen{1-\frac{\chi(p)}{p^s}}^{-1}=\prod_{\zeta^{f_p}=1}\bparen{1-\frac{\zeta}{p^s}}^{-g_p}.\tag{2.23}
\end{gather}
Now, recall that \(\prod_{\zeta^{f_p}=1}(1-\zeta X)=1-X^{f_p}\) in \(\C[X]\text{.}\) Therefore, equation (2.23) transforms to
\begin{gather}
\prod_{\chi\in H}\bparen{1-\frac{\chi(p)}{p^s}}^{-1}=\bparen{1-\frac{1}{p^{f_ps}}}^{-g_p}.\tag{2.24}
\end{gather}
Now, consider the case when \(p\mid n\text{.}\) Following the notations used in Lemma 2.17, let \(K'=K\cap\Q(\mu_{n'})\text{.}\) Then, by Lemma 2.17, \(K'\) is the maximal extension of \(\Q\) in \(K\) that is unramified at \(p\text{.}\) Let \(\mf{p}\) be a prime in \(K'\) lying above \(p\) and \(\mf{P}\) be a prime in \(K\) lying above \(\mf{p}\text{.}\) Then we have a tower of extensions \(K_{\mf{P}}/K'_{\mf{p}}/\Q_p\) local fields, with \([K_{\mf{P}}/\Q_p]=e_pf_p\text{.}\) Now, \(K'_{\mf{p}}/\Q_p\) is the unique maximal unramified extension in \(K_{\mf{P}}\) corresponding to the finite field extension \(\mathcal{O}_{K'}/\mf{p}/\F_p\) and hence \([K'_{p}:\Q_p]=f_p\text{.}\) Therefore, \([K_{\mf{P}}:K_{\mf{p}}]=e_p\text{,}\) which implies that \(K_{\mf{P}}/K'_{\mf{p}}\) is totally ramified. This is true for all primes \(\mf{p}\) of \(K'\) lying above \(p\text{.}\) As a result, if we pass from \(K\) to \(K'\text{,}\) the product in equation (2.22) does not change as \(f_p\) and \(g_p\) does not change. Moreover, whenever \(p\) divides the conductor of \(\chi\in H\text{,}\) \(\chi(p)=0\) and the Euler factor at \(p\) in equation (2.23) does not contribute anything. So, we pass to the subgroup \(H'\) of \(H\) of characters whose conductors are relatively prime to \(p\text{.}\) Therefore,
\begin{gather}
\prod_{\chi\in H}\bparen{1-\frac{\chi(p)}{p^s}}^{-1}=\prod_{\chi\in H'}\bparen{1-\frac{\chi(p)}{p^s}}^{-1}.\tag{2.25}
\end{gather}
Therefore, we are left to prove that \(K'=\Q(\mu_{n'})^{H'^{\perp}}\text{.}\) Note that \(H'=H\cap\mathcal{X}(n')\) by Lemma 2.17. By [47, Corollary 3.6], \(K'=\Q(\mu_n)^{H'^\perp}\text{.}\) Now, by Lemma 2.17,
\begin{gather}
K'=K'\cap\Q(\mu_{n'})=\Q(\mu_{n'})\cap\Q(\mu_n)^{H'\perp}=\Q(\mu_{n'})^{H'^\perp}.\tag{2.26}
\end{gather}
Now, since passing from \(K\) to \(K'\) does not change the product (2.22), passing from \(H\) to \(H'\) does not change the product on the right-hand side of equation (2.25), and \(K'=\Q(\mu_{n'})^{H'^\perp}\text{,}\) our argument in the first part of the proof applies as \(p\nmid n'\text{.}\) This completes the proof.
