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:

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

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

• Definition:Abscissa of Convergence
• Definition:Abscissa of Absolute Convergence

• Definition:General Dirichlet Series

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

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): L-Series
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Dirichlet series