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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 Extremal orders of an arithmetic function ： ウィキペディア英語版 Extremal orders of an arithmetic function In mathematics, in number theory, the extremal orders of an arithmetic function are best possible bounds of the given arithmetic function. Specifically, if ''f''(''n'') is an arithmetic function and ''m''(''n'') is a non-decreasing function that is ultimately positive and :$\backslash liminf\_\; \backslash frac\; =\; 1$ we say that ''m'' is a minimal order for ''f''. Similarly if ''M''(''n'') is a non-decreasing function that is ultimately positive and :$\backslash limsup\_\; \backslash frac\; =\; 1$ we say that ''M'' is a maximal order for ''f''.〔 〕 The subject was first studied systematically by Ramanujan starting in 1915.〔 ==Examples== * For the sum-of-divisors function σ(''n'') we have the trivial result ::$\backslash liminf\_\; \backslash frac\; =\; 1$ :because always σ(''n'') ≥ ''n'' and for primes σ(''p'') = ''p'' + 1. We also have ::$\backslash limsup\_\; \backslash frac\; =\; e^\backslash gamma,$ :proved by Gronwall in 1913.〔〔 〕 Therefore ''n'' is a minimal order and ''e''−γ ''n'' ln ln ''n'' is a maximal order for σ(''n''). * For the Euler totient φ(''n'') we have the trivial result ::$\backslash liminf\_\; \backslash frac\; =\; 1$ :because always φ(''n'') ≤ ''n'' and for primes φ(''p'') = ''p'' − 1. We also have ::$\backslash liminf\_\; \backslash frac\; =\; e^,$ :proved by Landau in 1903.〔〔 * For the number of divisors function ''d''(''n'') we have the trivial lower bound 2 ≤ ''d''(''n''), in which equality occurs when ''n'' is prime, so 2 is a minimal order. For ln ''d''(''n'') we have a maximal order ln 2 ln ''n'' / ln ln ''n'', proved by Wigert in 1907.〔〔 * For the number of distinct prime factors ω(''n'') we have a trivial lower bound 1 ≤ ω(''n''), in which equality occurs when ''n'' is a prime power. A maximal order for ω(''n'') is ln ''n'' / ln ln ''n''.〔 * For the number of prime factors counted with multiplicity Ω(''n'') we have a trivial lower bound 1 ≤ Ω(''n''), in which equality occurs when ''n'' is prime. A maximal order for Ω(''n'') is ln ''n'' / ln 2.〔 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「Extremal orders 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.