Dirichlet’s Theorem

This paper provides a proof of Dirichlet’s theorem, which states that when (m,a) = 1, there are infinitely many primes p such that p ≡ a (mod m). The proof explained here mirrors Serre’s proof in [1]. However, one distinction is that the Riemann zeta function is used as motivation to obtain results for the Dirichlet L-functions, which yields the above result. In fact, this paper can be read from the viewpoint of asking how do simple results of the zeta function depend on basic properties. For the rest of the paper, m is a fixed integer as given in the above theorem. Let P deonote the set of primes, Pm,a denote the set of primes that are congruent to a modulo m and Gm = (Z/mZ)×. We will use the Euler-phi function φ(m) which counts the numbers of positive integers less than m that are coprime to m. In particular, |Gm| = φ(m). A sequence of complex numbers {an} is called strictly multiplicative if an · am = anm,∀n,m ∈ N. The concept of analytic continuation will be useful in getting a deeper understanding of the proofs. However, prior knowledge is not necessary, and comments relating to analytic continuation can be ignored. Basic group theory is assumed. In particular, the result that abelian groups can be written as the product of cyclic groups is used without proof. 1 Riemann-zeta Function In this section, we recall and derive certain basic properties of the Riemann zeta function. For each s ∈ C,Re(s) > 1, the Riemann zeta function is defined as 1