# 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**.

## Also see

### Examples

- An ordinary Dirichlet series is the case when $\lambda_n = \log \left(n\right)$.
- Setting $\lambda_n = n$ gives a power series of exponential type.

## Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.

## Sources

- 1915: G.H. Hardy and Marcel Riesz:
*The General Theory of Dirichlet's Series*... (next): $\text I$: Introduction $\S 1$