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In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form : It can be resummed formally by expanding the denominator: : where the coefficients of the new series are given by the Dirichlet convolution of ''a''''n'' with the constant function 1(''n'') = 1: : 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 : where is the number of positive divisors of the number ''n''. For the higher order sigma functions, one has : where is any complex number and : 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 : : For Euler's totient function : : For Liouville's function : : 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
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