Definition:General Dirichlet Series

Definition

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

Let $\sequence {\lambda_n}$ 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:

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

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.

However, that makes it sound as though it is named after a high-ranking military officer.

Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.