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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 Prime-counting function ： ウィキペディア英語版 Prime-counting function In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by $\backslash scriptstyle\backslash pi(x)$ (unrelated to the number ). ==History== Of great interest in number theory is the growth rate of the prime-counting function.〔(【引用サイトリンク】 title=How many primes are there? )〕 It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately :$x/\backslash operatorname(x)\backslash !$ in the sense that :$\backslash lim\_\backslash frac=1.\backslash !$ This statement is the prime number theorem. An equivalent statement is :$\backslash lim\_\backslash pi(x)\; /\; \backslash operatorname(x)=1\backslash !$ where ''li'' is the logarithmic integral function. The prime number theorem was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function introduced by Riemann in 1859. More precise estimates of $\backslash pi(x)\backslash !$ are now known; for example :$\backslash pi(x)\; =\; \backslash operatorname(x)\; +\; O\backslash bigl(xe^\backslash bigr)\backslash !$ where the ''O'' is big O notation. For most values of $x$ we are interested in (i.e., when $x$ is not unreasonably large) $\backslash operatorname(x)\backslash !$ is greater than $\backslash pi(x)\backslash !$, but infinitely often the opposite is true. For a discussion of this, see Skewes' number. Proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by Atle Selberg and by Paul Erdős (for the most part independently). 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「Prime-counting 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.