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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 Bombieri–Vinogradov theorem ： ウィキペディア英語版 Bombieri–Vinogradov theorem In mathematics, the Bombieri–Vinogradov theorem (sometimes simply called Bombieri's theorem) is a major result of analytic number theory, obtained in the mid-1960s, concerning the distribution of primes in arithmetic progressions, averaged over a range of moduli. The first result of this kind was obtained by Mark Barban in 1961 and the Bombieri–Vinogradov theorem is a refinement of Barban's result. The Bombieri–Vinogradov theorem is named after Enrico Bombieri and A. I. Vinogradov,〔 Corrigendum. ibid. 30 (1966), pages 719-720. (Russian)〕 who published on a related topic, the density hypothesis, in 1965. Perhaps somewhat unjustly, Barban's name is often forgotten. This result is a major application of the large sieve method, which developed rapidly in the early 1960s, from its beginnings in work of Yuri Linnik two decades earlier. Besides Bombieri, Klaus Roth was working in this area. ==Statement of the Bombieri–Vinogradov theorem== Let $x$ and $Q$ be any two positive real numbers with :$x^\backslash log^x\backslash leq\; Q\backslash leq\; x^.$ Then :$\backslash sum\_\backslash max\_\backslash max\_\backslash left|\backslash psi(y;q,a)-\backslash right|=O\backslash left(x^Q(\backslash log\; x)^5\backslash right).$ Here $\backslash varphi(q)$ is the Euler totient function, which is the number of summands for the modulus ''q'', and :$\backslash psi(x;q,a)=\backslash sum\_\backslash Lambda(n),$ where $\backslash Lambda$ denotes the von Mangoldt function. A verbal description of this result is that it addresses the error term in the prime number theorem for arithmetic progressions, averaged over the moduli ''q'' up to ''Q''. For a certain range of ''Q'', which are around $\backslash sqrt\; x$ if we neglect logarithmic factors, the error averaged is nearly as small as $\backslash sqrt\; x$. This is not obvious, and without the averaging is about of the strength of the Generalized Riemann Hypothesis (GRH). 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「Bombieri–Vinogradov theorem」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.