Prime Numbers in Arithmetic Progressions

Abstract

The method of complex integration together with the results of L-functions proved in Chapter VIII will enable us to write down an explicit formula connecting the sum of values of the function Λ(n) over the integers lying in a given arithmetic progression with the zeros of an L-function. This explicit formula together with a theorem on the boundary of the zeros of the L-function will yield the prime number theorem for arithmetic progressions. We shall always assume below that k ≤ x.