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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 hypothesis for curves over finite fields ： ウィキペディア英語版 Local zeta-function Suppose that ''V'' is a non-singular ''n''-dimensional projective algebraic variety over the field F''q'' with ''q'' elements. In number theory, the local zeta function ''Z''(''V'', ''s'') of ''V'' (sometimes called the congruent zeta function) is defined as :$Z(V,\; s)\; =\; \backslash exp\backslash left(\backslash sum\_^\backslash infty\; \backslash frac\; (q^)^m\backslash right)$ where ''N''''m'' is the number of points of ''V'' defined over the degree ''m'' extension F''q''''m'' of F''q''. By the variable transformation $u=q^$, then it is defined by :$$ \mathit (V,u) = \exp \left( \sum_^ N_m \frac \right) as the formal power series of the variable ''u''. Equivalently, sometimes it is defined as follows: :$$ (1)\ \ \mathit (V,0) = 1 \, :$$ (2)\ \ \frac \log \mathit (V,u) = \sum_^ N_m u^\ . In other word, the local zeta function ''Z(V,u)'' with coefficients in the finite field F is defined as a function whose logarithmic derivative generates the numbers ''Nm'' of the solutions of equation, defining ''V'', in the ''m'' degree extension F''m''. ==Formulation== Given ''F'', there is, up to isomorphism, just one field ''Fk'' with :$(F\_k\; :\; F)\; =\; k\; \backslash ,$, for ''k'' = 1, 2, ... . Given a set of polynomial equations — or an algebraic variety ''V'' — defined over ''F'', we can count the number :$N\_k\; \backslash ,$ of solutions in ''Fk'' and create the generating function :$G(t)\; =\; N\_1t\; +N\_2t^2/2\; +\; N\_3t^3/3\; +\backslash cdots\; \backslash ,$. The correct definition for ''Z''(''t'') is to make log ''Z'' equal to ''G'', and so :$Z=\; \backslash exp\; (G(t))\; \backslash ,$ we will have ''Z''(0) = 1 since ''G''(0) = 0, and ''Z''(''t'') is ''a priori'' a formal power series. Note that the logarithmic derivative :$Z\text{'}(t)/Z(t)\; \backslash ,$ equals the generating function :$G\text{'}(t)\; =\; N\_1\; +N\_2t^1\; +\; N\_3t^2\; +\backslash cdots\; \backslash ,$. 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「Local 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.