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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 arithmetic function ： ウィキペディア英語版 arithmetic function In number theory, an arithmetic, arithmetical, or number-theoretic function is a real or complex valued function ''f''(''n'') defined on the set of natural numbers (i.e. positive integers) that "expresses some arithmetical property of ''n''".〔Hardy & Wright, intro. to Ch. XVI〕 An example of an arithmetic function is the non-principal character (mod 4) defined by :$$ \chi(n) = \left(\frac\right)= \begin \;\;\,0 & \text n \text, \\ \;\;\, 1 & \text n \equiv 1 \mod 4, \\ -1 & \text n \equiv 3 \mod 4, \end     where $(\backslash tfrac)\backslash$ is the Kronecker symbol. To emphasize that they are being thought of as functions rather than sequences, values of an arithmetic function are usually denoted by ''a''(''n'') rather than ''a''''n''. There is a larger class of number-theoretic functions that do not fit the above definition, e.g. the prime-counting functions. This article provides links to functions of both classes. ==Notation== $\backslash sum\_p\; f(p)\backslash ;$   and   $\backslash prod\_p\; f(p)\backslash ;$   mean that the sum or product is over all prime numbers: :$\backslash sum\_p\; f(p)\; =\; f(2)\; +\; f(3)\; +\; f(5)\; +\; \backslash cdots$     $\backslash prod\_p\; f(p)=\; f(2)f(3)f(5)\backslash cdots.$ Similarly,   $\backslash sum\_\; f(p^k)\backslash ;$   and   $\backslash prod\_\; f(p^k)\backslash ;$   mean that the sum or product is over all prime powers with strictly positive exponent (so 1 is not included): :$\backslash sum\_\; f(p^k)\; =\; f(2)\; +\; f(3)\; +\; f(4)\; +f(5)\; +f(7)+f(8)+f(9)+\backslash cdots$ $\backslash sum\_\; f(d)\backslash ;$   and   $\backslash prod\_\; f(d)\backslash ;$   mean that the sum or product is over all positive divisors of ''n'', including 1 and ''n''. E.g., if ''n'' = 12, :$\backslash prod\_\; f(d)\; =\; f(1)f(2)\; f(3)\; f(4)\; f(6)\; f(12).\backslash$ The notations can be combined:   $\backslash sum\_\; f(p)\backslash ;$   and   $\backslash prod\_\; f(p)\backslash ;$   mean that the sum or product is over all prime divisors of ''n''. E.g., if ''n'' = 18, :$\backslash sum\_\; f(p)\; =\; f(2)\; +\; f(3),\backslash$ and similarly   $\backslash sum\_\; f(p^k)\backslash ;$   and   $\backslash prod\_\; f(p^k)\backslash ;$   mean that the sum or product is over all prime powers dividing ''n''. E.g., if ''n'' = 24, :$\backslash prod\_\; f(p^k)\; =\; f(2)\; f(3)\; f(4)\; f(8).\backslash$ 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「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.