Editing Series (mathematics) (section)

===Dirichlet series===
:{{Main|Dirichlet series}}

A [[Dirichlet series]] is one of the form

<math display=block>\sum_{n=1}^\infty {a_n \over n^s},</math>

where {{tmath|s}} is a [[complex number]].  For example, if all {{tmath|a_n}} are equal to {{tmath|1}}, then the Dirichlet series is the [[Riemann zeta function]]

<math display=block>\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.</math>

Like the zeta function, Dirichlet series in general play an important role in [[analytic number theory]].  Generally a Dirichlet series converges if the real part of {{tmath|s}} is greater than a number called the abscissa of convergence.  In many cases, a Dirichlet series can be extended to an [[analytic function]] outside the domain of convergence by [[analytic continuation]].  For example, the Dirichlet series for the zeta function converges absolutely when {{tmath|\operatorname{Re}(s)>1}}, but the zeta function can be extended to a holomorphic function defined on <math>\Complex\setminus\{1\}</math> with a simple [[pole (complex analysis)|pole]] at&nbsp;{{tmath|1}}.

This series can be directly generalized to [[general Dirichlet series]].