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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 Prime zeta function ： ウィキペディア英語版 Prime zeta function In mathematics, the Prime zeta function is an analogue of the Riemann zeta function, studied by . It is defined as the following infinite series, which converges for $\backslash Re(s)\; >\; 1$: :$P(s)=\backslash sum\_$. == Properties == The Euler product for the Riemann zeta function ζ(''s'') implies that :$\backslash log\backslash zeta(s)=\backslash sum\_\; \backslash frac$ which by Möbius inversion gives :$P(s)=\backslash sum\_\; \backslash mu(n)\backslash frac$ When ''s'' goes to 1, we have $P(s)\backslash sim\; \backslash log\backslash zeta(s)\backslash sim\backslash log\backslash left(\backslash frac\backslash right)$. This is used in the definition of Dirichlet density. This gives the continuation of ''P''(''s'') to $\backslash Re(s)\; >\; 0$, with an infinite number of logarithmic singularities at points ''s'' where ''ns'' is a pole (only ''ns=1'')), or zero of the Riemann zeta function ζ(.). The line $\backslash Re(s)\; =\; 0$ is a natural boundary as the singularities cluster near all points of this line. If we define a sequence :$a\_n=\backslash prod\_\; \backslash frac=\backslash prod\_\; \backslash frac$ then :$P(s)=\backslash log\backslash sum\_^\backslash infty\; \backslash frac.$ (Exponentiation shows that this is equivalent to Lemma 2.7 by Li.) The prime zeta function is related with the Artin's constant by :$\backslash ln\; C\_^\; \backslash frac$ where ''L''''n'' is the ''n''th Lucas number. Specific values are: + \tfrac + \tfrac + \tfrac + \tfrac + \cdots \to \infty. || |- | 2 || $045224\backslash text74200\backslash text41065\backslash text49850\; \backslash ldots$ || |- | 3 || $017476\backslash text26392\backslash text99443\backslash text53642\; \backslash ldots$ || |- | 4 || $007699\backslash text31397\backslash text64246\backslash text84494\; \backslash ldots$ || |- | 5 || $003575\backslash text50174\backslash text83924\backslash text25713\; \backslash ldots$ || |- | 9 || $000200\backslash text44675\backslash text74962\backslash text45066\; \backslash ldots$ || |} 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「Prime 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.