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.