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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 Average order of an arithmetic function ： ウィキペディア英語版 Average order of an arithmetic function In number theory, an average order of an arithmetic function is some simpler or better-understood function which takes the same values "on average". Let be an arithmetic function. We say that an ''average order'' of is if :$\backslash sum\_\; f(n)\; \backslash sim\; \backslash sum\_\; g(n)$ as tends to infinity. It is conventional to choose an approximating function that is continuous and monotone. But even so an average order is of course not unique. In cases where the limit : $\backslash lim\_\backslash frac\backslash sum\_\; f(n)=c$ exists, it is said that has a mean value (average value) . ==Examples== * An average order of , the number of divisors of , is ; * An average order of , the sum of divisors of , is ; * An average order of , Euler's totient function of , is ; * An average order of , the number of ways of expressing as a sum of two squares, is ; * The average order of representations of a natural number as a sum of three squares is ; * The average number of decompositions of a natural number into a sum of one or more consecutive prime numbers is ; * An average order of , the number of distinct prime factors of , is ; * An average order of , the number of prime factors of , is ; * The prime number theorem is equivalent to the statement that the von Mangoldt function has average order 1; * An average order of , the Möbius function, is zero; this is again equivalent to the prime number theorem. ==Calculating mean values using Dirichlet series== In case ''F'' is of the form : $F(n)=\backslash sum\_\; f(d),$ for some arithmetic function ''f''(''n''), one has, : $\backslash sum\_\; F(n)=\backslash sum\_\; f(d)\; \backslash sum\_\; 1=\backslash sum\_\; f(d)()\; =\; x\backslash sum\_\; \backslash frac\; \backslash text\; +\; O(\backslash sum\_\; |f(d)|).\backslash qquad\backslash qquad\; (1)$ This identity often provides a practical way to calculate the mean value in terms of the Riemann zeta function. This is illustrated in the following example. 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「Average order of an arithmetic 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.