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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 Apéry's theorem ： ウィキペディア英語版 Apéry's theorem In mathematics, Apéry's theorem is a result in number theory that states the Apéry's constant ζ(3) is irrational. That is, the number :$\backslash zeta(3)=\backslash sum\_^\backslash infty\; \backslash frac=\backslash frac+\backslash frac+\backslash frac+\backslash ldots\; =\; 1.2020569\backslash ldots$ cannot be written as a fraction ''p''/''q'' with ''p'' and ''q'' being integers. The special values of Riemann zeta function at even integers ''2n'' (''n > 0'') can be shown as terms of Bernoulli numbers and then to be irrational as follows while it remains open whether they are in general rational or not at the odd integers ''2n + 1'' (''n > 0'') though they are conjectured to be irrational. ==History== Euler has proven that if ''n'' is a positive integer then :$\backslash frac\backslash pi^$ for some rational number ''p''/''q''. Specifically, writing the infinite series on the left as ζ(2''n'') he showed :$\backslash zeta(2n)\; =\; (-1)^\backslash frac\}$ where the ''Bn'' are the rational Bernoulli numbers. Once it was proved that π''n'' is always irrational this showed that ζ(2''n'') is irrational for all positive integers ''n''. No such representation in terms of π is known for the so-called zeta constants for odd arguments, the values ζ(2''n''+1) for positive integers ''n''. It has been conjectured that the ratios of these quantities :$\backslash frac\{\backslash pi^\{2n+1\}\},$ are transcendental for every integer ''n'' ≥ 1. Because of this, no proof could be found to show that the zeta constants with odd arguments were irrational, even though they were—and still are—all believed to be transcendental. However, in June 1978, Roger Apéry gave a talk entitled "Sur l'irrationalité de ζ(3)." During the course of the talk he outlined proofs that ζ(3) and ζ(2) were irrational, the latter using methods simplified from those used to tackle the former rather than relying on the expression in terms of π. Due to the wholly unexpected nature of the result and Apéry's blasé and very sketchy approach to the subject many of the mathematicians in the audience dismissed the proof as flawed. However Henri Cohen, Hendrik Lenstra, and Alfred van der Poorten suspected Apéry was onto something and set out to confirm his proof. Two months later they finished verification of Apéry's proof, and on August the 18th Cohen delivered a lecture giving full details of the proof. After the lecture Apéry himself took to the podium to explain the source of some of his ideas. 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「Apéry's theorem」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.