A Catalog of Interesting Dirichlet Series

Abstract

A Dirichlet series is a series of the form $$F(s)=\sum_{n=1}^\infty {f(n)\over n^s},$$ where the variable $s$ may be complex or real and $f(n)$ is a number-theoretic function. The sum of the series, $F(s)$, is called the generating function of $f(n)$. The Riemann zeta-function $$\zeta(s)=\sum_{n=1}^{\infty}{1\over n^{s}}=\Pi_{p}\left(1-{1\over p^{s}}\right)^{-1},$$ where $n$ runs through all integers and $p$ runs through all primes is the special case where $f(n) =1$ identically. It is fundamental to the study of prime numbers and many generating functions are combinations of this function. In this paper, we give an overview of some of the commonly known number-theoretic functions together with their corresponding Dirichlet series.