Definition:Dirichlet Series

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


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

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

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

Examples

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