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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 Legendre's constant ： ウィキペディア英語版 Legendre's constant Legendre's constant is a mathematical constant occurring in a formula conjectured by Adrien-Marie Legendre to capture the asymptotic behavior of the prime-counting function $\backslash scriptstyle\backslash pi(x)$. Its value is now known to be exactly 1. Examination of available numerical evidence for known primes led Legendre to suspect that $\backslash scriptstyle\backslash pi(x)$ satisfies an approximate formula. Legendre conjectured in 1808 that :$\backslash pi(x)\; =\; \backslash frac$ where $\backslash lim\_\; B(x)\; =\; 1.08366$.... Or similarly, :$\backslash lim\_\; \backslash left(\; \backslash ln(n)\; -\; \backslash right)=\; B$ where ''B'' is Legendre's constant. He guessed ''B'' to be about 1.08366, but regardless of its exact value, the existence of ''B'' implies the prime number theorem. Pafnuty Chebyshev proved in 1849〔Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen, page 17. Third (corrected) edition, two volumes in one, 1974, Chelsea 1974〕 that if the limit ''B'' exists, it must be equal to 1. An easier proof was given by Pintz in 1980.〔J. Pintz. On Legendre's prime number formula. Amer. Math. Monthly 87 (1980), 733-735.〕 It is an immediate consequence of the prime number theorem, under the precise form with an explicit estimate of the error term :$\backslash pi(x)=\; (x)\; +\; O\; \backslash left(x\; \backslash mathrm^\; x\; \backslash to\; \backslash infty$ (for some positive constant ''a'', where ''O''(…) is the big O notation), as proved in 1899 by Charles de La Vallée Poussin,〔La Vallée Poussin, C. Mém. Couronnés Acad. Roy. Belgique 59, 1-74, 1899〕 that ''B'' indeed is equal to 1. (The prime number theorem had been proved in 1896, independently by Jacques Hadamard〔''Sur la distribution des zéros de la fonction $\backslash zeta(s)$ et ses conséquences arithmétiques'', Bulletin de la Société Mathématique de France, Vol. 24, 1896, pp. 199–220 (Online )〕 and La Vallée Poussin,〔« Recherches analytiques sur la théorie des nombres premiers », Annales de la société scientifique de Bruxelles, vol. 20, 1896, p. 183-256 et 281-361〕 but without any estimate of the involved error term). Being evaluated to such a simple number has made the term Legendre's constant mostly only of historical value, with it often (technically incorrectly) being used to refer to Legendre's first guess 1.08366... instead. Pierre Dusart proved in 2010 : for $x\; \backslash ge\; 5393$, and : for $x\; \backslash ge\; 60184$. This is of the same form as :$\backslash pi(x)\; =\; \backslash frac$ with . ==References== 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「Legendre's constant」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.