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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 Euler product ： ウィキペディア英語版 Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The name arose from the case of the Riemann zeta-function, where such a product representation was proved by Leonhard Euler. ==Definition== In general, if $a$ is a multiplicative function, then the Dirichlet series :$\backslash sum\_\; a(n)n^\backslash ,$ is equal to :$\backslash prod\_\; P(p,\; s)\backslash ,$ where the product is taken over prime numbers $p$, and $P(p,\; s)$ is the sum :$1+a(p)p^\; +\; a(p^2)p^\; +\; \backslash cdots\; .$ In fact, if we consider these as formal generating functions, the existence of such a ''formal'' Euler product expansion is a necessary and sufficient condition that $a(n)$ be multiplicative: this says exactly that $a(n)$ is the product of the $a(p^k)$ whenever $n$ factors as the product of the powers $p^k$ of distinct primes $p$. An important special case is that in which $a(n)$ is totally multiplicative, so that $P(p,\; s)$ is a geometric series. Then :$P(p,\; s)=\backslash frac\{1-a(p)p^\{-s\}\},$ as is the case for the Riemann zeta-function, where $a(n)\; =\; 1$, and more generally for Dirichlet characters. 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「Euler product」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.