Definition:Dirichlet Series

Definition
A Dirichlet series is function $f:\C \to \C$ defined by a series:


 * $\displaystyle f(s) = \sum_{n=1}^\infty { a_n n^{-s} }$

where $s\in \C$ and $a_n: \N \to \C$ is an arithmetic function.

It is a historical convention that the variable $s$ is written $s=\sigma + it$ with $\sigma,t\in \R$.

Examples

 * The Riemann zeta function is the Dirichlet series with $a_n = 1$ for all $n$