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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 Möbius inversion ： ウィキペディア英語版 Möbius inversion formula In mathematics, the classic Möbius inversion formula was introduced into number theory during the 19th century by August Ferdinand Möbius. Other Möbius inversion formulas are obtained when different locally finite partially ordered sets replace the classic case of the natural numbers ordered by divisibility; for an account of those, see incidence algebra. ==Statement of the formula== The classic version states that if ''g'' and ''f'' are arithmetic functions satisfying : $g(n)=\backslash sum\_f(d)\backslash quad\backslash textn\backslash ge\; 1$ then :$f(n)=\backslash sum\_\backslash mu(d)g(n/d)\backslash quad\backslash textn\backslash ge\; 1$ where μ is the Möbius function and the sums extend over all positive divisors ''d'' of ''n''. In effect, the original ''f''(''n'') can be determined given ''g''(''n'') by using the inversion formula. The two sequences are said to be Möbius transforms of each other. The formula is also correct if ''f'' and ''g'' are functions from the positive integers into some abelian group (viewed as a $\backslash mathbb$-module). In the language of Dirichlet convolutions, the first formula may be written as :$g=f$ *1 where '' *'' denotes the Dirichlet convolution, and ''1'' is the constant function $1(n)=1$. The second formula is then written as :$f=\backslash mu$ * g. Many specific examples are given in the article on multiplicative functions. The theorem follows because $$ * is (commutative and) associative, and $1$ * \mu = \epsilon, where $\backslash epsilon$ is the identity function for the Dirichlet convolution, taking values $\backslash epsilon(1)\; =\; 1,\; \backslash epsilon(n)\; =\; 0$ for all $n\; >\; 1$. Thus $\backslash mu$ * g = \mu * (1 * f) = (\mu * 1) * f = \epsilon * f = f. 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「Möbius inversion formula」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.