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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 Chebyshev's function ： ウィキペディア英語版 Chebyshev function In mathematics, the Chebyshev function is either of two related functions. The first Chebyshev function ''ϑ''(''x'') or ''θ''(''x'') is given by :$\backslash vartheta(x)=\backslash sum\_\; \backslash log\; p$ with the sum extending over all prime numbers ''p'' that are less than or equal to ''x''. The second Chebyshev function ''ψ''(''x'') is defined similarly, with the sum extending over all prime powers not exceeding ''x'': :$\backslash psi(x)\; =\; \backslash sum\_\backslash log\; p=\backslash sum\_\; \backslash Lambda(n)\; =\; \backslash sum\_\backslash lfloor\backslash log\_p\; x\backslash rfloor\backslash log\; p,$ where $\backslash Lambda$ is the von Mangoldt function. The Chebyshev functions, especially the second one ''ψ''(''x''), are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, ''π''(''x'') (See the exact formula, below.) Both Chebyshev functions are asymptotic to ''x'', a statement equivalent to the prime number theorem. Both functions are named in honour of Pafnuty Chebyshev. ==Relationships== The second Chebyshev function can be seen to be related to the first by writing it as :$\backslash psi(x)=\backslash sum\_\; k\; \backslash log\; p$ where ''k'' is the unique integer such that ''p''''k'' ≤ ''x'' and ''x'' < ''p''''k''+1. The values ''k'' of are given in . A more direct relationship is given by :$\backslash psi(x)=\backslash sum\_^\backslash infty\; \backslash vartheta\; \backslash left(x^\backslash right).$ Note that this last sum has only a finite number of non-vanishing terms, as :$\backslash vartheta\; \backslash left(x^\backslash right)\; =\; 0\backslash textn>\backslash log\_2\; x\backslash \; =\; \backslash frac,.$ The second Chebyshev function is the logarithm of the least common multiple of the integers from 1 to ''n''. :$\backslash operatorname(1,2,\backslash dots,\; n)=e^.$ Values of  $\backslash operatorname(1,2,\backslash dots,\; n)$  for the integer variable ''n'' is given at . 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「Chebyshev 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.