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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 Riemann-Siegel formula ： ウィキペディア英語版 Riemann–Siegel formula In mathematics, the Riemann–Siegel formula is an asymptotic formula for the error of the approximate functional equation of the Riemann zeta function, an approximation of the zeta function by a sum of two finite Dirichlet series. It was found by in unpublished manuscripts of Bernhard Riemann dating from the 1850s. Siegel derived it from the Riemann–Siegel integral formula, an expression for the zeta function involving contour integrals. It is often used to compute values of the Riemann–Siegel formula, sometimes in combination with the Odlyzko–Schönhage algorithm which speeds it up considerably. When used along the critical line, it is often useful to use it in a form where it becomes a formula for the Z function. If ''M'' and ''N'' are non-negative integers, then the zeta function is equal to :$\backslash zeta(s)\; =\; \backslash sum\_^N\backslash frac\; +\; \backslash gamma(1-s)\backslash sum\_^M\backslash frac-s\}\; \backslash frac\; \backslash right)\}(1-s)\backslash right)\}$ is the factor appearing in the functional equation , and :$R(s)\; =\; \backslash frac\backslash int\; \backslash frac\}dx$ is a contour integral whose contour starts and ends at +∞ and circles the singularities of absolute value at most . The approximate functional equation gives an estimate for the size of the error term. and derive the Riemann–Siegel formula from this by applying the method of steepest descent to this integral to give an asymptotic expansion for the error term ''R''(''s'') as a series of negative powers of Im(''s''). In applications ''s'' is usually on the critical line, and the positive integers ''M'' and ''N'' are chosen to be about . found good bounds for the error of the Riemann–Siegel formula. ==Riemann's integral formula== Riemann showed that :$\backslash int\_\; \backslash frac-e^\}\; du\; =\; \backslash frac\}\}$ where the contour of integration is a line of slope −1 passing between 0 and 1 . He used this to give the following integral formula for the zeta function: :$\backslash pi^\}\backslash Gamma\backslash left\; (\backslash tfrac\; \backslash right)\backslash zeta(s)=\; \backslash pi^\}\backslash Gamma\; \backslash left\; (\backslash tfrac\; \backslash right)\; \backslash int\_\backslash frac\}\}\backslash ,dx\; +\backslash pi^\}\backslash Gamma\; \backslash left\; (\backslash tfrac\; \backslash right)\backslash int\_\backslash frac\}\}\backslash ,dx$ 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「Riemann–Siegel formula」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.