The Dirichlet Series and the Dirichlet Theorem on Primes in Arithmetic Progressions

Abstract

In the last two chapters, we studied $\zeta (s)$ given by the series $\displaystyle {\sum ^{\infty }_{n=1} \ \frac{1}{n^s}}$ in $\sigma > 1.$ In this chapter, we consider more general series $\displaystyle {\sum ^{\infty }_{n=1} \ \frac{a_n}{n^s}}$ with $a_n\in \mathbf{C}$ and $s \in \mathbf{C}$ and these are known as the Dirichlet series.