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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 Dedekind psi function ： ウィキペディア英語版 Dedekind psi function In number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by :$\backslash psi(n)\; =\; n\; \backslash prod\_\backslash left(1+\backslash frac\backslash right),$ where the product is taken over all primes p dividing n (by convention, ψ(1) is the empty product and so has value 1). The function was introduced by Richard Dedekind in connection with modular functions. The value of ψ(''n'') for the first few integers ''n'' is: :1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24 ... . ψ(''n'') is greater than ''n'' for all ''n'' greater than 1, and is even for all ''n'' greater than 2. If ''n'' is a square-free number then ψ(''n'') = σ(''n''). The ψ function can also be defined by setting ψ(pn) = (p+1)pn-1 for powers of any prime ''p'', and then extending the definition to all integers by multiplicativity. This also leads to a proof of the generating function in terms of the Riemann zeta function, which is :$\backslash sum\; \backslash frac\; =\; \backslash frac.$ This is also a consequence of the fact that we can write as a Dirichlet convolution of $\backslash psi=\; \backslash mathrm$ * |\mu| . ==Higher Orders== The generalization to higher orders via ratios of Jordan's totient is :$\backslash psi\_k(n)=\backslash frac$ with Dirichlet series :$\backslash sum\_\backslash frac\; =\; \backslash frac$. It is also the Dirichlet convolution of a power and the square of the Möbius function, :$\backslash psi\_k(n)\; =\; n^k$ * \mu^2(n). If :$\backslash epsilon\_2\; =\; 1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0\backslash ldots$ is the characteristic function of the squares, another Dirichlet convolution leads to the generalized σ-function, :$\backslash epsilon\_2(n)$ * \psi_k(n) = \sigma_k(n). 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「Dedekind psi 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.