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:''For the rational functions defined on the complex numbers, see Möbius transformation.'' The classical Möbius function ''μ''(''n'') is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832.〔Hardy & Wright, Notes on ch. XVI: "... μ(''n'') occurs implicitly in the works of Euler as early as 1748, but Möbius, in 1832, was the first to investigate its properties systematically."〕〔In the ''Disquisitiones Arithmeticae'' (1801) Carl Friedrich Gauss showed that the sum of the primitive roots (mod ''p'') is μ(''p'' − 1), (see #Properties and applications) but he didn't make further use of the function. In particular, he didn't use Möbius inversion in the ''Disquisitiones''.〕 It is a special case of a more general object in combinatorics. ==Definition== For any positive integer ''n'', define ''μ''(''n'') as the sum of the . It has values in * ''μ''(''n'') = 1 if ''n'' is a square-free positive integer with an even number of prime factors. * ''μ''(''n'') = −1 if ''n'' is a square-free positive integer with an odd number of prime factors. * ''μ''(''n'') = 0 if ''n'' has a squared prime factor. The values of ''μ''(''n'') for the first 30 positive numbers are The first 50 values of the function are plotted below: 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 function」の詳細全文を読む スポンサード リンク
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