Dirichlet L-functions

Abstract

The theory of analytic functions has many applications in number theory. A particularly spectacular application was discovered by Dirichlet who proved in 1837 that there are infinitely many primes in any arithmetic progression b, b + m, b + 2m,..., where (m, b) = 1. To do this he introduced the L functions which bear his name. In this chapter we will define these functions, investigate their properties, and prove the theorem on arithmetic pro­gressions. The use of Dirichlet L-functions extends beyond the proof of this theorem. It turns out that their values at negative integers are especially important. We will derive these values and show how they relate to Bernoulli numbers.