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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 (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 : in the sense that : This statement is the prime number theorem. An equivalent statement is : 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 are now known; for example : where the ''O'' is big O notation. For most values of we are interested in (i.e., when is not unreasonably large) is greater than , 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
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