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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 Dirichlet eta function ： ウィキペディア英語版 Dirichlet eta function In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: :$\backslash eta(s)\; =\; \backslash sum\_^\; =\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots$ This Dirichlet series is the alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function, ζ(s) — and for this reason the Dirichlet eta function is also known as the alternating zeta function, also denoted ζ *(s). The following simple relation holds: :$\backslash eta(s)\; =\; \backslash left(1-2^\backslash right)\; \backslash zeta(s)$ While the Dirichlet series expansion for the eta function is convergent only for any complex number ''s'' with real part > 0, it is Abel summable for any complex number. This serves to define the eta function as an entire function (and the above relation then shows the zeta function is meromorphic with a simple pole at ''s'' = 1, and perhaps poles at the other zeros of the factor $1-2^$). Equivalently, we may begin by defining :$\backslash eta(s)\; =\; \backslash frac\backslash int\_0^\backslash infty\; \backslash frac$ which is also defined in the region of positive real part. This gives the eta function as a Mellin transform. Hardy gave a simple proof of the functional equation for the eta function, which is :$\backslash eta(-s)\; =\; 2\; \backslash frac\}\; \backslash pi^\; s\; \backslash sin\backslash left(\backslash right)\; \backslash Gamma(s)\backslash eta(s+1).$ From this, one immediately has the functional equation of the zeta function also, as well as another means to extend the definition of eta to the entire complex plane. ==Zeros== The zeros of the eta function include all the zeros of the zeta function: the infinity of negative even integers (real equidistant simple zeros); an infinity of zeros along the critical line, none of which are known to be multiple and over 40% of which have been proven to be simple, and the hypothetical zeros in the critical strip but not on the critical line, which if they do exist must occur at the vertices of rectangles symmetrical around the ''x''-axis and the critical line and whose multiplicity is unknown. In addition, the factor $1-2^$ adds an infinity of complex simple zeros, located at equidistant points on the line $\backslash Re(s)=1$, at $s\_n=1+2n\backslash pi\; i/\backslash log(2)$ where ''n'' is any nonzero integer. Under the Riemann hypothesis, the zeros of the eta function would be located symmetrically with respect to the real axis on two parallel lines $\backslash Re(s)=1/2,\; \backslash Re(s)=1$, and on the perpendicular half line formed by the negative real axis. 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「Dirichlet eta 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.