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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 series ： ウィキペディア英語版 Dirichlet series In mathematics, a Dirichlet series is any series of the form :$\backslash sum\_^\; \backslash frac,$ where ''s'' is complex, and ''a'' is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Peter Gustav Lejeune Dirichlet. ==Combinatorial importance== Dirichlet series can be used as generating series for counting weighted sets of objects with respect to a weight which is combined multiplicatively when taking Cartesian products. Suppose that ''A'' is a set with a function ''w'': ''A'' → N assigning a weight to each of the elements of ''A'', and suppose additionally that the fibre over any natural number under that weight is a finite set. (We call such an arrangement (''A'',''w'') a weighted set.) Suppose additionally that ''an'' is the number of elements of ''A'' with weight ''n''. Then we define the formal Dirichlet generating series for ''A'' with respect to ''w'' as follows: :$\backslash mathfrak^A\_w(s)\; =\; \backslash sum\_\; \backslash frac\; =\; \backslash sum\_^\; \backslash frac$ Note that if ''A'' and ''B'' are disjoint subsets of some weighted set (''U'', ''w''), then the Dirichlet series for their (disjoint) union is equal to the sum of their Dirichlet series: :$\backslash mathfrak^\_w(s)\; =\; \backslash mathfrak^\_w(s)\; +\; \backslash mathfrak^\_w(s).$ Moreover, and perhaps a bit more interestingly, if (''A'', ''u'') and (''B'', ''v'') are two weighted sets, and we define a weight function ''w'': ''A'' × ''B'' → N by :$w(a,b)\; =\; u(a)\; v(b),$ for all ''a'' in ''A'' and ''b'' in ''B'', then we have the following decomposition for the Dirichlet series of the Cartesian product: :$\backslash mathfrak^\_w(s)\; =\; \backslash mathfrak^\_u(s)\; \backslash cdot\; \backslash mathfrak^\_v(s).$ This follows ultimately from the simple fact that $n^\; \backslash cdot\; m^\; =\; (nm)^.$ 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 series」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.