# 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:

- $\displaystyle \map f s = \sum_{n \mathop = 1}^\infty a_n n^{-s}$

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 sources use the term **Dirichlet $L$-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 $\sigma, t \in \R$.

## Also see

### Examples

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

## Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Dirichlet series**