Analytic number theory

3 Zeta functions & The class number formula

Given a number field \(K\), let \(j_K(n)\) be the number of ideals of norm \(n\) in \(\cO _K\). The Dedekind zeta function \(\zeta _K(s)\) is the Dirichlet series for \(j_K(n)\)

Lemma 3.1. \(\zeta _K(s)\) converges for \(\real (s)>1\).

Proof. From the theorem on the distribution of ideals, it follows that \(\sum _{i\leq n}j_K(i)=O(n)\), so that by Lemma 2.2 this is true. □

We will want to extend \(\zeta _K(s)\) to \(\real (s)>1-\frac {1}{[K:\QQ ]}\), so that we can better study its behavior near \(1\). First \(\zeta (s)\) will be extended. Note that \(\zeta (s)(1-2^{1-s})\) is the Dirichlet series for the sequence \((-1)^{n+1}\), which converges for \(\real (s)>0\). This lets us extend \(\zeta (s)\) to \(\real (s)>0\), but it might have some singularities where \(1-2^{1-s}\) is \(0\). To reduce the number of possible poles, \(\zeta (s)(1-3^{1-s})\) also converges for \(\real (s)>0\) for the same reason, but the only common zeros of \(1-2^{1-s}\) and \(1-3^{1-s}\) are \(s =1\). Thus \(\zeta (s)\) can only have a simple pole at \(s =1\), and indeed it does since \(1-2^{1-s}\) has a simple zero. Now for some \(\kappa \), \(\zeta _K(s)-\kappa \zeta (s)\) is the Dirichlet series for some sequence \(f(n)\) with \(\sum _1^n f(i) = O(n^{1-\frac 1{[K:\QQ ]}})\) so we can use this to extend \(\zeta _K(s)\). Note that as a consequence, \(\zeta _K(s)\) also has a simple pole at \(1\).

Theorem 3.2. \(\zeta _K(s)\) is an analytic function in the region \(\real (s)>1-\frac {1}{[K:\QQ ]}\), and has a simple pole with residue \(\frac {h_K2^{r+s}\pi ^s \reg (K)}{\omega _K \sqrt {|\disc (K)|}}\).

Proof. Because of how we have shown that \(\zeta _K(s)\) is analytic near \(1\), we only need to compute the residue of \(\zeta (s)\) at \(0\). Note that the Dirichlet series for \((-1)^{s+1}\) at \(1\) is \(\sum _i \frac {(-1)^{i+1}}{i} = \int _0^1\sum _i (-1)^{i+1}x^{i+1} = \int _0^1\frac {1}{1+x} = \log (2)\). On the other hand, the \(s-1\) term in the series expansion of \((1-2^{1-s})\) is \(\frac {1}{\log (2)}\). □

Now we can write \(\zeta _K(s)\) another way. But first a technical lemma.

Lemma 3.3. Suppose that \(a_i \in \CC \) satisfy \(|a_i|<1\) and \(\sum _i |a_i| \leq \infty \). Then \(\prod _1^\infty (1-a_i)^{-1} = \sum _{S \subset \NN \hspace {.1cm}\text {finite}}\prod _S a_i\), where the sum is absolutely convergent.

Proof. Note that \(e^{-\sum _i |a_i|} = \prod _i e^{-|a_i|}\) converges and by Taylor’s theorem is at least \(\prod _i(1-|a_i|)\). Thus the inverse, which is the left hand side, absolutely converges, and the partial products show that it converges to the right hand side. □

Proposition 3.4. For \(\real (s)>1\), \(\zeta _K(s) = \sum _{I \subset \cO _K}\frac {1}{N(I)} = \prod _{\pp \subset \cO _K}\frac {1}{1-N(\pp )^{-s}}\).

Proof. These follow from Lemma 3.3, the fact that \(\cO _K\) is a Dedekind domain, and the fact that \(\zeta _K(s)\) converges in that region. □

For any subset of primes \(S\), we can also consider \(\zeta _{K,S}(s) = \prod _{\pp \subset S}\frac {1}{1-N(\pp )^{-s}}\). Note that it converges for \(\real (s)>1\) as well.

Lemma 3.5. Let \(S\) be a set of primes in \(\cO _K\). Then \(|\sum _{\pp \in S} \frac {1}{N(\pp )^s} -\log (\zeta _{K,S}(s))| = O(1)\) for \(\real (s)>1\).

Proof. \(\log (\zeta _K(s))=\sum _{\pp \in S} -\log (1-N(\pp )^s) = \sum _{\pp \in S} \frac {1}{N(\pp ^s)} + \sum _{\pp \in S,m\geq 2} \frac {1}{m\pp ^{ms}}\).The second term is at most \(\sum _{\pp ,m\geq 2} \frac {1}{N(\pp )^{ms}} = \sum _{\pp \in S} \frac {1}{N(\pp )^{2s}-N(\pp )^{s}} \leq 2\sum _{\pp \in S} N(\pp )^{-2}<\infty \). □

The Dirichlet density of a set of primes \(S\) is the limit as \(s \to 1\), if it exists, of \(\frac {\sum _{\pp \subset S} \frac {1}{N(\pp )^s}}{\sum _{\pp } \frac {1}{N(\pp )^s}}\). We can similarly define the upper and lower Dirichlet densities using \(\liminf \) and \(\limsup \). It clearly satisfies natural properties one might expect. We say that \(S\) has polar density \(\frac {n}{m}\) if \(\zeta _{K,S}(s)^m\) has a pole of order \(n\). Polar density also satisfies natural properties. The following is easy to see because of Lemma 3.5.

Lemma 3.6. If the polar density exists, so does the Dirichlet density, and the two are equal.

Lemma 3.7. Let \(A\) be a set of primes with residual degree \(> 1\) over \(\ZZ \). Then \(\zeta _{K,A}(s)\) is holomorphic for \(\real (s)>\frac {1}{2}\) and \(A\) has polar density \(0\).

Proof. Split \(A\) up by residual degree \(f\), and note that the terms corresponding zeta function is bounded by \(\zeta _K(fs)^n\), so by Lemma 2.2 each is holomorphic on \(\real (s)>\frac {1}{f}\). □

Corollary 3.8. The primes in \(K\) that split completely in a Galois extension \(L\) have polar density \(\frac {1}{[M:K]}\) where \(M\) is the Galois closure of \(L/K\).

Proof. First assume \(L\) is Galois. WLOG, we can restrict to primes that have residual degree \(1\) over \(\QQ \). But then this is clear, since if \(A\) are the primes of residual degree \(1\) in \(L\) and \(B\) are the primes of residual degree \(1\) splitting completely, then \(\zeta _{L,A}(s)\) is the \([L:K]^{th}\) power of the \(\zeta _{K,B}(s)\), so the result follows. Now we can remove the Galois assumption by noting that a prime splits completely in \(L\) iff it splits completely in the Galois closure. □

Corollary 3.9. Let \(H\) be a subgroup of \(\ZZ /m\ZZ ^\times \). Then the primes congruent to \(H\) have polar density \(\frac {|H|}{m}\).

Proof. Apply the previous result to the corresponding abelian extension of \(\QQ \). □

A Dirichlet character mod \(n\) (usually denoted \(\chi \)) is a totally multiplicative function on the natural numbers factoring through \(\ZZ /n\ZZ \), and supported on \(\ZZ /n\ZZ ^\times \). The character corresponding to the trivial homomorphism is called the trivial character, and denoted \(1\). We can define \(L(s,\chi ) = \sum _1^\infty \frac {\chi (n)}{n^s}\) to be the Dirichlet \(L\)-series.

Lemma 3.10. \(L(s,\chi )\) is holomorphic on \(\real (s)>0\) when \(\chi \) is nontrivial, and has a simple pole when \(\chi \) is trivial.

Proof. Note that \(\sum _{\ZZ /n\ZZ } \chi (a) = 0\) for a nontrivial character mod \(n\), so that \(\sum _1^m \chi (a) = O(1)\), giving the first result. \(L(s,1) = \zeta (s)\prod _{p|n}(1- p^{-s})\), giving the second result. □

More generally, given an abelian extension \(L\) of a number field \(K\), we can consider characters \(\chi \in \hat {G}\) on the Galois group \(G\), and we can define a corresponding character on the set of ideals via the Artin map composed with the character. For simplicity, let \(\chi (I)\) be shorthand for \(\chi ((\frac {L/K}{I}))\). Then \(L_K(s,\chi ) = \prod _{\pp \in \Spec (\cO _K)^{ur}}(1-\frac {\chi (\pp )}{\pp ^{s}}) = \sum _{I \in I^{ur}}\frac {\chi (I)}{N(I)^s}\).

Proposition 3.11. Let \(f_\pp \) be the residual degree of every prime over \(\pp \) in \(L\), and let \(r_\pp \) be the number of factors \(\pp \) splits into. \(\zeta _L(s) = \prod _{\pp \not \in \Spec (\cO _K)^{ur}}(1-N(\pp )^{-f_\pp s})^{r_\pp } \prod _{\chi \in \hat {G}}L(s,\chi )\). \(\zeta _K(s,1) = \zeta _{K,\Spec (\cO _K)^{ur}}(s)\).

Proof. The last identity is obvious, so we’ll focus on the first. We can split the factors of \((1-N(\mf {P})^{-s})^{-1}\) on the left hand side into Galois orbits. For each prime \(\pp \), we will get \(\prod _{\mf {P}/\pp }(1-\frac {1}{N(\mf {P})^{s}})^{-r_\pp }\), and for the unramified primes, this factors into \(\prod _{\chi \in \hat {G}}(1-\frac {\chi (\pp )}{N(\pp )^{s}})^{-1}\) by basic facts about characters of abelian groups, and how Frobenius relates to splitting. □

Corollary 3.12. If \(\chi \) is a nontrivial Dirichlet character, then \(L(1,\chi ) \neq 0\).

Proof. This follows from the above proposition, the fact that \(L(s,\chi )\) are holomorphic at \(1\) for nontrivial \(\chi \), and the fact that \(L(s,1),\zeta _{\QQ [\zeta _m]}(s)\) have a simple pole at \(1\). □

From the proposition, it follows that if \(K\) is a subfield of a cyclotomic field (i.e. any abelian extension) where primes dividing \(m\) ramify, then the residue at \(1\) of \(\zeta _K(s)\) is also given by \(\prod _{p\nmid m}(1-\frac {1}{p^{f_p}})^{r_p}(1-\frac {1}{p})^{-1} \prod _{1\neq \chi \in \hat {G}}L(1,\chi )\), giving a class number formula. To compute this, we will need to evaluate \(L(1,\chi )\) for nontrivial characters.

Theorem 3.13. Let \(\chi \) be a nontrivial Dirichlet character mod \(m\). Then \(L(1,\chi ) = -\frac {1}{m}\sum _1^{m-1}\tau _k(\chi )\log (1-\omega ^{-k}))\) where \(\tau _k(\chi )\) is the Gauss sum \(\sum _{\ZZ /m\ZZ ^\times }\chi (a)\omega ^{ak}\), and \(\omega \) a primitive \(m^{th}\) root of unity.

Proof. \(L(s,\chi ) = \sum _a \frac {\chi (a)}{a^s} = \sum _{b \in \ZZ /m\ZZ ^\times }\chi (b)\sum _{a \equiv b \pmod m} \frac {\chi (a)}{a^s}\) \(=\frac {1}{m}\sum _{b \in \ZZ /m\ZZ ^\times }\chi (b)\sum _{a} \frac {\sum _0^{m-1}\omega ^{(b-a)k}}{a^s} = \frac {1}{m}\sum _1^{m-1}\tau _k(\chi )\sum _{a}\frac {\omega ^{-ak}}{a^s}\), and evaluating at \(s = 1\) gives the result. □

We can simplify this formula as follows: If \(\chi \) is a character mod \(m\) and is not induced from one mod \(n \neq m\), then it is primitive. If \(\chi \) mod \(m\) is induced from \(\chi '\) mod \(n\), then we have the formula \(L(s,\chi ) = L(s,\chi ')\prod _{p|m,p\nmid n}(1-\frac {\chi '(p)}{p^s})\), and every character is induced from a primitive one, so we only need to be able to compute \(L(s,\chi )\) for primitive characters. Let \(\tau (\chi ) = \tau _1(\chi )\). Then for any character mod \(m\), if \((k,m) = 1\), it is easy to see \(\tau _k(\chi ) = \overline {\chi }(k)\tau (\chi )\). More generally, \(\tau _k(\chi ) = \chi (1+i\frac {m}{(m,k)})\tau _k(\chi )\), and so if \(\chi \) is primitive and \((k,m)>1\), then we have \(\tau _k(\chi ) = 0\).

Thus \(L(1,\chi ) = -\frac {\tau (\chi )}{m}\sum _{\ZZ /m\ZZ ^\times }\overline {\chi }(k)\log (1-\omega ^{-k}))\).