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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Rational zeta series ： ウィキペディア英語版 Rational zeta series In mathematics, a rational zeta series is the representation of an arbitrary real number in terms of a series consisting of rational numbers and the Riemann zeta function or the Hurwitz zeta function. Specifically, given a real number ''x'', the rational zeta series for ''x'' is given by :$x=\backslash sum\_^\backslash infty\; q\_n\; \backslash zeta\; (n,m)$ where ''q''''n'' is a rational number, the value ''m'' is held fixed, and ζ(''s'', ''m'') is the Hurwitz zeta function. It is not hard to show that any real number ''x'' can be expanded in this way. ==Elementary series== For integer ''m>1'', one has :$x=\backslash sum\_^\backslash infty\; q\_n\; \backslash left(\backslash sum\_^\; k^\backslash right)$ For ''m=2'', a number of interesting numbers have a simple expression as rational zeta series: :$1=\backslash sum\_^\backslash infty\; \backslash left()$ and :$1-\backslash gamma=\backslash sum\_^\backslash infty\; \backslash frac\backslash left()$ where γ is the Euler–Mascheroni constant. The series :$\backslash log\; 2\; =\backslash sum\_^\backslash infty\; \backslash frac\backslash left()$ follows by summing the Gauss–Kuzmin distribution. There are also series for π: :$\backslash log\; \backslash pi\; =\backslash sum\_^\backslash infty\; \backslash frac\backslash left()$ and :$\backslash frac\; -\; \backslash frac\; =\backslash sum\_^\backslash infty\; \backslash frac^\backslash infty\; (-1)^\; t^\; \backslash left()\; =$ \frac + \frac - \frac which in turn follows from the generating function for the Bernoulli numbers :$\backslash frac\; =\; \backslash sum\_^\backslash infty\; B\_n\; \backslash frac$ Adamchik and Srivastava give a similar series :$\backslash sum\_^\backslash infty\; \backslash frac\; \backslash zeta(2n)\; =$ \log \left(\frac \right) 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「Rational zeta series」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.