Definition:Dirichlet Series

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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


Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.


Sources