Dirichlet’s theorem on primes in arithmetic progressions (digression)

So. First off, surely the five remaining readers of this increasingly erratic blog have noticed the change of theme.

I want to next discuss the second inequality in class field theory, which is an upper bound on the norm index of the idele group. There are two ways I know of to prove this: one analytic, one algebraic. I will first sketch the analytic one. I say sketch because to do a full proof would get into the details of Dirichlet series, lattice points in homogeneously expanding domains, the construction of a certain fundamental domain for the action of the units, etc., etc., and I’d rather outline those ideas rather than do all the details because this is a series on class field theory. What I do plan on doing properly, however, is the algebraic (due to Chevalley in 1940) proof of the second inequality, which heavily uses results of field theory (e.g. Kummer theory) and local fields (e.g. power index computations). I still thought it worthwhile to sketch the analytic approach, though. Rather than jumping right into it (I have to first say something about how the ideal and idele groups are connected), I decided to give an expository post on L-functions and Dirichlet’s theorem—in the case of the rational numbers.

1. Ramblings on the Riemann-zeta function

Recall that the Riemann-zeta function is defined by ${\zeta(s) = \sum n^{-s}}$ , and that it is intimately connected with the distribution of the prime numbers because of the product formula

$\displaystyle \zeta(s) = \prod_p (1 - p^{-s})^{-1}$

valid for ${Re(s)>1}$ , and which is a simple example of unique factorization. In particular, we have

$\displaystyle \log \zeta(s) = \sum_p p^{-s} + O(1) , \ s \rightarrow 1^+.$

It is known that ${\zeta(s)}$ has an analytic continuation to the whole plane with a simple pole with residue one at ${1}$ . The easiest way to see this is to construct the analytic continuation for ${Re (s)>0}$ . For instance, ${\zeta(s) - \frac{1}{s-1}}$ can be represented as a certain integral for ${Re(s)>1}$ that actually converges for ${Re(s)>0}$ though. (The functional equation is then used for the rest of the analytic continuation.) The details are here for instance. As a corollary, it follows that

$\displaystyle \sum_p p^{-s} = \log \frac{1}{s-1} + O(1) , \ s \rightarrow 1^+.$

This fact can be used in deducing properties about the prime numbers. (Maybe sometime I’ll discuss the proof of the prime number theorem on this blog.) Much simpler than that, however, is the proof of Dirichlet’s theorem on the infinitude of primes in arithmetic progressions. I will briefly outline the proof of this theorem, since it will motivate the idea of L-functions.

Theorem 1 (Dirichlet) Let ${\{an+b\}_{n \in \mathbb{Z}}}$ be an arithmetic progression with ${a,b}$ relatively prime. Then it contains infinitely many primes.

The idea of this proof is to note that the elements of the arithmetic progression ${\{an+b\}}$ can be characterized by so-called “Dirichlet characters.” This is actually a general and very useful (though technically trivial) fact about abelian groups, which I will describe now.

2. Characters

Let ${G}$ be an abelian group. A character of ${G}$ is a homomorphism ${\chi: G \rightarrow \mathbb{C}^*}$ ; the characters themselves form a group, called the dual ${G^{\vee}}$ . It is a general fact that

$\displaystyle \sum_{g \in G} \chi(g) = 0$

unless ${\chi \equiv 1 }$ (in which case the sum is obviously ${|G|}$ ). We can rephrase this in another form. Consider the vector space ${Fun{G}}$ of complex-valued functions from ${G \rightarrow \mathbb{C}}$ . This space has a Hermitian inner product ${(\dot, \dot)}$ defined via ${(f,f') = \frac{1}{|G|} \sum_{g \in G} f(g) \overline{f'(g)}}$ . So, in particular, we see that the characters of ${G}$ form an orthonormal set with respect to this inner product. By Fourier theory, they are actually an orthonormal basis! But, we don’t need Fourier theory for this. We can directly show that there are precisely ${|G|}$ characters. Indeed, ${G}$ is a direct sum of cyclic groups, and it is easy to check that there is a noncanonical isomorphism ${G \simeq G^{\vee}}$ for ${G}$ cyclic (and hence for ${G}$ finite abelian). Abstract nonsensical aside: Although the isomorphism ${G \rightarrow G^{\vee}}$ is noncanonical, the isomorphism between ${G}$ and ${(G^{\vee})^{\vee}}$ is actually canonical. This is similar to the situation in Eilenberg and Maclane’s paper where category theory began. The key fact we aim to prove is that:

Proposition 2 If ${a \in G}$ , then the function $\displaystyle g \rightarrow \sum_{\chi} \chi(a)^{-1} \chi(g)$

is equal to the characteristic function of ${\{a\}}$ .

This is now simply the fact that any ${f \in Fun{G}}$ has a Fourier expansion ${f = \sum_{\chi} (f, \chi) \chi}$ applied to the characteristic function of ${a}$ ! Although this is true only for ${G}$ abelian, there is still something very interesting that holds for ${G}$ nonabelian. One has to look not simply at group-homorphisms into ${\mathbb{C}^{\vee}}$ , but group-homomorphisms into linear groups ${GL_n(\mathbb{C})}$ , i.e. group representations. A character is obtained by taking the trace of such a group-homomorphism. It turns out that the irreducible characters form an orthonormal basis for the subspace of ${Fun{G}}$ of functions constant on conjugacy classes. All this is covered in any basic text on representation theory of finite groups, e.g. Serre’s.

3. Dirichlet’s theorem and L-functions

The way to prove Dirichlet’s theorem is to consider sums of the form

$\displaystyle \sum_{p \equiv b \ \mod a} p^{-s}$

and to prove they are unbounded as ${s \rightarrow 1^+}$ . But, this is badly behaved because of the additivity in the notion of congruence. Here is what one does to accomodate the intrinsically multiplicative nature of the primes. Fix ${a,b}$ now as in the statement of Dirichlet’s theorem. By the previous section, we now know the identity

$\displaystyle \sum_{p \equiv b \ \mod a} p^{-s} = \frac{1}{\phi(a)}\sum_{\chi} \chi(b)^{-1} \sum_p \chi(p) p^{-s}$

where ${\chi}$ ranges over the characters of ${(\mathbb{Z}/a \mathbb{Z})^{*}}$ , extended to ${\mathbb{N}}$ as functions taking the value zero one numbers not prime to ${a}$ . The beauty of this is that the sum on the left, which before involved the ugly additive notion of congruence, has been replaced by sums of the form

$\displaystyle \Phi(\chi, s)= \sum_p \chi(p) p^{-s},$

and the function ${\chi}$ is multiplicative. This in fact looks a lot like our expression for ${\log \zeta}$ , but with ${\chi}$ terms introduced. In fact, this leads us to introduce the L-function

$\displaystyle L(s, \chi) = \sum \chi(n) n^{-s}.$

It follows similarly using the multiplicativity of ${\chi}$ and unique factorization that we have a product formula

$\displaystyle L(s, \chi) = \prod_p (1 - \chi(p) p^{-s})^{-1}$

and in particular, ${\Phi(\chi, s) \sim \log L(s, \chi)}$ . It thus becomes crucial to study the behavior of ${L}$ -functions as ${s \rightarrow 1^+}$ . When ${\chi}$ is the unit character, then ${L(s, \chi)}$ is ${\prod_{p \mid a} (1 - p^{-s})}$ which is basically the zeta function itself, so ${\log L(s, \chi) \simeq \log \frac{1}{s-1}}$ . The first thing to notice is that analytic continuation is much easier for L-functions when ${\chi}$ is nontrivial. We have by summation by parts:

$\displaystyle \sum n^{-s} \chi(n) = \sum (n^{-s} - (n-1)^{s}) \left( \sum_{k \leq n} \chi(k) \right),$

and the last term is ${\leq C n^{-s-1}}$ because the character sum is bounded—this follows because ${\sum_{k} \chi(k)=0}$ when ${k}$ ranges over a representatives of residue classes modulo ${a}$ . This thus converges whenever ${Re(s)>0}$ . Since we have

$\displaystyle \sum_{p \equiv b \ \mod a} p^{-s} \sim \frac{1}{\phi(b)} \log\frac{1}{s-1} + \frac{1}{\phi(b)} \sum_{\chi \neq 1} \chi(b)^{-1} \log L(s, \chi),$

we will need to know whether ${L(1, \chi) = 0}$ for nontrivial ${\chi}$ . If this is not the case, then the sum is unbounded as ${s \rightarrow 1^+}$ , and in particular there must be infinitely many primes congruent to ${b}$ modulo ${a}$ . In particular, Dirichlet’s theorem follows from:

Theorem 3 If ${\chi \not\equiv 1}$ , then ${L(1, \chi) \neq 0}$ .

4. Nonvanishing of the L-series

I will prove this theorem (i.e. ${L(1, \chi) \neq 0}$ for ${\chi \not\equiv 1}$ ) using a nifty trick that I learned a while back in Serre’s A Course in Arithmetic, forgot, googled, and re-found in these notes of Pete Clark. LINK The trick is to consider the product

$\displaystyle \zeta_N(s) = \zeta(s) \prod_{\chi \neq 1} L(s, \chi)$

which by analytic continuation, is at least a meromorphic function in ${Re(s)>0}$ with at most a pole of order 1 at ${s=1}$ . If any one of ${L(1, \chi)=0}$ , then ${\zeta_N}$ is actually analytic in the entire half-plane. But, let us look at what this product looks like: it is

$\displaystyle \prod_p \prod_{\chi} ( 1 - \chi(p) p^{-s})^{-1}$

where ${\chi = 1}$ is allowed. Suppose ${p}$ has order ${f}$ modulo ${a}$ . Then ${\chi(p)}$ ranges over the ${f}$ -th roots of unity, taking each one ${\phi(a)/f}$ times, as ${\chi}$ ranges over the characters of ${a}$ . By the identity ${\prod_{\zeta^k = 1} (1 - X \zeta) = X^k -1}$ , we find

$\displaystyle \zeta_N(s) = \prod_p ( 1- p^{-fs})^{-\phi(a)/f}.$

This holds for ${s>1}$ . This product, however, is bigger than

$\displaystyle \prod_p ( 1- p^{-\phi(a)s})$

for ${s>1}$ . This last product can be expanded as a Dirichlet series, however, when ${s \rightarrow \frac{1}{\phi(a)}}$ it blows up to ${\infty}$ since ${\prod_p(1-p^{-1})}$ diverges to zero. Using facts about Dirichlet series, it follows that ${\zeta_N(s)}$ must have a pole on the positive real axis, which is impossible if it is analytic in the entire half-plane. Bingo.