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In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function—which is obtained by specializing to the case where ''K'' is the rational numbers Q. In particular, it can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at ''s'' = 1, and its values encode arithmetic data of ''K''. The extended Riemann hypothesis states that if ''ζ''''K''(''s'') = 0 and 0 < Re(''s'') < 1, then Re(''s'') = 1/2. The Dedekind zeta function is named for Richard Dedekind who introduced them in his supplement to Peter Gustav Lejeune Dirichlet's Vorlesungen über Zahlentheorie. ==Definition and basic properties== Let ''K'' be an algebraic number field. Its Dedekind zeta function is first defined for complex numbers ''s'' with real part Re(''s'') > 1 by the Dirichlet series : where ''I'' ranges through the non-zero ideals of the ring of integers ''O''''K'' of ''K'' and ''N''''K''/Q(''I'') denotes the absolute norm of ''I'' (which is equal to both the index () of ''I'' in ''O''''K'' or equivalently the cardinality of quotient ring ''O''''K'' / ''I''). This sum converges absolutely for all complex numbers ''s'' with real part Re(''s'') > 1. In the case ''K'' = Q, this definition reduces to that of the Riemann zeta function. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dedekind zeta function」の詳細全文を読む スポンサード リンク
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