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:


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

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 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 $n$.