An Elementary Proof of Dirichlet's Theorem About Primes in an Arithmetic Progression

Abstract

1. A classical theorem by Dirichlet asserts that every arithmetic progression ky + 1, where the positive integers k and 1 are relatively prime, represents infinitely many primes as y runs over the positive integers. The object of this paper is to give a new and more elementary proof of this theorem. More elementary in the respect that we do not use the complex characters mod k, and also in that we consider only finite sums. More precisely the theorem that is proved in this paper is the following: For every positive integer k, there exist positive numbers Ck and xo depending only on k, such that, when (k, 1) = 1 we have