Dirichlet’s Theorem on Primes in an Arithmetic Progression

Our goal is to prove the following theorem: Dirichlet’s Theorem: For any coprime a, b ∈ Z, there are infinitely many primes p such that p ≡ a (mod b). Although the statement of the theorem involves only integers, the simplest proof requires the use of complex numbers and Dirichlet L-series. Most of this paper will therefore be devoted to proving some basic properties of characters and L-series, after which the desired theorem will follow fairly easily. The main sources for the following proof are Daniel Marcus’ Number Fields and Chapter 7 from Chan Heng Huat’s online course notes from MA 4263: Analytic Number Theory at the National University of Singapore (available at http://www.math.nus.edu.sg/ ̃chanhh/MA4263/MA4263.html), especially the latter. Marcus proves a more general result, while Chan’s proof is a little easier to follow. In addition, our proof that ∑