Editing Dedekind zeta function (section)

{{short description|Generalization of the Riemann zeta function for algebraic number fields}}
In [[mathematics]], the '''Dedekind zeta function''' of an [[algebraic number field]] ''K'', generally denoted ζ<sub>''K''</sub>(''s''), is a generalization of the [[Riemann zeta function]] (which is obtained in the case where ''K'' is the [[rational number|field of rational numbers]] '''Q'''). It can be defined as a [[Dirichlet series]], it has an [[Euler product]] expansion, it satisfies a [[functional equation (L-function)|functional equation]], it has an [[analytic continuation]] to a [[meromorphic function]] on the [[complex plane]] '''C''' with only a [[simple pole]] at ''s''&nbsp;=&nbsp;1, and its values encode arithmetic data of ''K''. The [[extended Riemann hypothesis]] states that if ''ζ''<sub>''K''</sub>(''s'')&nbsp;=&nbsp;0 and 0&nbsp;<&nbsp;Re(''s'')&nbsp;<&nbsp;1, then Re(''s'')&nbsp;=&nbsp;1/2.

The Dedekind zeta function is named for [[Richard Dedekind]] who introduced it in his supplement to [[Peter Gustav Lejeune Dirichlet]]'s [[Vorlesungen über Zahlentheorie]].<ref>{{harvnb|Narkiewicz|2004|loc=§7.4.1}}</ref>