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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 Z function ： ウィキペディア英語版 Z function In mathematics, the Z-function is a function used for studying the Riemann zeta-function along the critical line where the real part of the argument is one-half. It is also called the Riemann-Siegel Z-function, the Riemann-Siegel zeta-function, the Hardy function, the Hardy Z-function and the Hardy zeta-function. It can be defined in terms of the Riemann-Siegel theta-function and the Riemann zeta-function by :$Z(t)\; =\; e^\; \backslash zeta\backslash left(\backslash frac+it\backslash right).$ It follows from the functional equation of the Riemann zeta-function that the Z-function is real for real values of ''t''. It is an even function, and real analytic for real values. It follows from the fact that the Riemann-Siegel theta-function and the Riemann zeta-function are both holomorphic in the critical strip, where the imaginary part of ''t'' is between -1/2 and 1/2, that the Z-function is holomorphic in the critical strip also. Moreover, the real zeros of Z(''t'') are precisely the zeros of the zeta-function along the critical line, and complex zeros in the Z-function critical strip correspond to zeros off the critical line of the Riemann zeta-function in its critical strip. ==The Riemann-Siegel formula== Calculation of the value of Z(t) for real t, and hence of the zeta-function along the critical line, is greatly expedited by the Riemann–Siegel formula. This formula tells us :$Z(t)\; =\; 2\; \backslash sum\_\; n^\backslash cos(\backslash theta(t)-t\; \backslash log\; n)\; +R(t),$ where the error term R(t) has a complex asymptotic expression in terms of the function :$\backslash Psi(z)\; =\; \backslash frac$ and its derivatives. If $u=(\backslash frac)^$,$N=\backslash lfloor\; u^2\; \backslash rfloor$ and $p\; =\; u^2\; -\; N$ then :$R(t)\; \backslash sim\; (-1)^$ \left( \Psi(p)u^ - \frac\Psi^(p)u^ + \cdots\right) where the ellipsis indicates we may continue on to higher and increasingly complex terms. Other efficient series for Z(t) are known, in particular several using the incomplete gamma function. If :$Q(a,\; z)\; =\; \backslash frac\; =\; \backslash frac\; \backslash int\_z^\backslash infty\; u^\; e^\; du$ then an especially nice example is :$Z(t)\; =2\; \backslash Re\; \backslash left(e^$ \left(\sum_^\infty Q\left(\frac,\pi i n^2 \right) - \frac} \right)} \right)\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「Z function」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.