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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 Selberg zeta function ： ウィキペディア英語版 Selberg zeta function The Selberg zeta-function was introduced by . It is analogous to the famous Riemann zeta function :$\backslash zeta(s)\; =\; \backslash prod\_$ is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics instead of the primes numbers.　If $\backslash Gamma$ is a subgroup of SL(2,R) Selberg zeta function is defined as follows, :$\backslash zeta\_\backslash Gamma(s)=\backslash prod\_p(1-N(p)^)^,$ or :$Z\_\backslash Gamma(s)=\backslash prod\_p\backslash prod^\backslash infty\_(1-N(p)^),$ where p run all over the prime congruent class and N(p) is the norm of congruent class p, which is square of the bigger eigenvalue of p. For any hyperbolic surface of finite area there is an associated Selberg zeta-function; this function is a meromorphic function defined in the complex plane. The zeta function is defined in terms of the closed geodesics of the surface. The zeros and poles of the Selberg zeta-function, ''Z''(''s''), can be described in terms of spectral data of the surface. The zeros are at the following points: # For every cusp form with eigenvalue $s\_0(1-s\_0)$ there exists a zero at the point $s\_0$. The order of the zero equals the dimension of the corresponding eigenspace. (A cusp form is an eigenfunction to the Laplace–Beltrami operator which has Fourier expansion with zero constant term.) # The zeta-function also has a zero at every pole of the determinant of the scattering matrix, $\backslash phi(s)$. The order of the zero equals the order of the corresponding pole of the scattering matrix. The zeta-function also has poles at $1/2\; -\; \backslash mathbb$, and can have zeros or poles at the points $-\; \backslash mathbb$. The Ihara zeta function is considered a p-adic (and a graph-theoretic) analogue of the Selberg zeta function. == Selberg zeta-function for the modular group == For the case where the surface is $\backslash Gamma\; \backslash backslash\; \backslash mathbb^2$, where $\backslash Gamma$ is the modular group, the Selberg zeta-function is of special interest. For this special case the Selberg zeta-function is intimately connected to the Riemann zeta-function. In this case the determinant of the scattering matrix is given by: :$\backslash varphi(s)\; =\; \backslash pi^\; \backslash frac.$ In particular, we see that if the Riemann zeta-function has a zero at $s\_0$, then the determinant of the scattering matrix has a pole at $s\_0/2$, and hence the Selberg zeta-function has a zero at $s\_0/2$. 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「Selberg zeta 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.