Editing Dirichlet L-function (section)

{{Short description|Type of mathematical function}}
{{DISPLAYTITLE:Dirichlet ''L''-function}}
In [[mathematics]], a '''Dirichlet''' <math>L</math>-'''series''' is a function of the form

:<math>L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}.</math>

where <math> \chi </math> is a [[Dirichlet character]] and <math> s </math> a [[complex variable]] with [[real part]] greater than <math> 1 </math>. It is a special case of a [[Dirichlet series]]. By [[analytic continuation]], it can be extended to a [[meromorphic function]] on the whole [[complex plane]], and is then called a '''Dirichlet <math> L </math>-function''' and also denoted <math> L ( s , \chi) </math>.

These functions are named after [[Peter Gustav Lejeune Dirichlet]] who introduced them in {{harv|Dirichlet|1837}} to prove the [[Dirichlet's theorem on arithmetic progressions|theorem on primes in arithmetic progressions]] that also bears his name. In the course of the proof, Dirichlet shows that <math> L ( s , \chi) </math> is non-zero at <math> s = 1 </math>. Moreover, if <math> \chi </math> is principal, then the corresponding Dirichlet <math> L </math>-function has a [[simple pole]] at <math> s = 1 </math>. Otherwise, the <math> L </math>-function is [[entire function|entire]].