# Definition:General Dirichlet Series

## Definition

Let $a_n$ be a sequence in $\C$.

Let $\left\langle{\lambda_n}\right\rangle$ be a strictly increasing sequence of non-negative real numbers whose limit is infinity.

A general Dirichlet series of type $\lambda_n$ is a complex function $f: \C \to \C$ defined by the series:

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

which is defined at the points where it converges.

## Notation

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

## Also known as

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

## Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.