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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 Lambert series ： ウィキペディア英語版 Lambert series In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form :$S(q)=\backslash sum\_^\backslash infty\; a\_n\; \backslash frac\; .$ It can be resummed formally by expanding the denominator: :$S(q)=\backslash sum\_^\backslash infty\; a\_n\; \backslash sum\_^\backslash infty\; q^\; =\; \backslash sum\_^\backslash infty\; b\_m\; q^m$ where the coefficients of the new series are given by the Dirichlet convolution of ''a''''n'' with the constant function 1(''n'') = 1: :$b\_m\; =\; (a$ *1)(m) = \sum_ a_n. \, This series may be inverted by means of the Möbius inversion formula, and is an example of a Möbius transform. ==Examples== Since this last sum is a typical number-theoretic sum, almost any natural multiplicative function will be exactly summable when used in a Lambert series. Thus, for example, one has :$\backslash sum\_^\backslash infty\; q^n\; \backslash sigma\_0(n)\; =\; \backslash sum\_^\backslash infty\; \backslash frac$ where $\backslash sigma\_0(n)=d(n)$ is the number of positive divisors of the number ''n''. For the higher order sigma functions, one has :$\backslash sum\_^\backslash infty\; q^n\; \backslash sigma\_\backslash alpha(n)\; =\; \backslash sum\_^\backslash infty\; \backslash frac$ where $\backslash alpha$ is any complex number and :$\backslash sigma\_\backslash alpha(n)\; =\; (\backslash textrm\_\backslash alpha$ *1)(n) = \sum_ d^\alpha \, is the divisor function. Lambert series in which the ''a''''n'' are trigonometric functions, for example, ''a''''n'' = sin(2''n'' ''x''), can be evaluated by various combinations of the logarithmic derivatives of Jacobi theta functions. Other Lambert series include those for the Möbius function $\backslash mu(n)$: :$\backslash sum\_^\backslash infty\; \backslash mu(n)\backslash ,\backslash frac\; =\; q.$ For Euler's totient function $\backslash phi(n)$: :$\backslash sum\_^\backslash infty\; \backslash varphi(n)\backslash ,\backslash frac\; =\; \backslash frac.$ For Liouville's function $\backslash lambda(n)$: :$\backslash sum\_^\backslash infty\; \backslash lambda(n)\backslash ,\backslash frac\; =$ \sum_^\infty q^ with the sum on the left similar to the Ramanujan theta function. 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「Lambert 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.