# Definition:Dirichlet Series

## Definition

Let $a_n: \N \to \C$ be an arithmetic function.

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

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

which is defined at the points where it converges.

## Also known as

A Dirichlet series is also known as an ordinary Dirichlet series to distinguish it from a general Dirichlet series.

Some treatments of this subject use the possessive style: Dirichlet's series.

## Notation

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

## Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.