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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 Jurkat–Richert theorem ： ウィキペディア英語版 Jurkat–Richert theorem The Jurkat–Richert theorem is a mathematical theorem in sieve theory. It is a key ingredient in proofs of Chen's theorem on Goldbach's conjecture.〔 〕 It was proved in 1965 by Wolfgang B. Jurkat and Hans-Egon Richert.〔 〕 ==Statement of the theorem== This formulation is from Diamond & Halberstam.〔 〕 Other formulations are in Jurkat & Richert,〔 Halberstam & Richert,〔 〕 and Nathanson.〔 Suppose ''A'' is a finite sequence of integers and ''P'' is a set of primes. Write ''A''''d'' for the number of items in ''A'' that are divisible by ''d'', and write ''P''(''z'') for the product of the elements in ''P'' that are less than ''z''. Write ω(''d'') for a multiplicative function such that ω(''p'')/''p'' is approximately the proportion of elements of ''A'' divisible by ''p'', write ''X'' for any convenient approximation to |''A''|, and write the remainder as : $r\_A\; (d)\; =\; \backslash left|\; A\_d\; \backslash right|\; -\; \backslash frac\; X.$ Write ''S''(''A'',''P'',''z'') for the number of items in ''A'' that are relatively prime to ''P''(''z''). Write : $V(z)\; =\; \backslash prod\_\; \backslash left(\; 1\; -\; \backslash frac\; \backslash right).$ Write ν(''m'') for the number of distinct prime divisors of ''m''. Write ''F''1 and ''f''1 for functions satisfying certain difference differential equations (see Diamond & Halberstam〔 for the definition and properties). We assume the dimension (sifting density) is 1: that is, there is a constant ''C'' such that for 2 ≤ ''z'' < ''w'' we have : $\backslash prod\_\; \backslash left(\; 1\; -\; \backslash frac\; \backslash right)^\; \backslash le\; \backslash left(\; \backslash frac\; \backslash right)\; \backslash left(\; 1\; +\; \backslash frac\; \backslash right).$ (The book of Diamond & Halberstam〔 extends the theorem to dimensions higher than 1.) Then the Jurkat–Richert theorem states that for any numbers ''y'' and ''z'' with 2 ≤ ''z'' ≤ ''y'' ≤ ''X'' we have : $S(A,P,z)\; \backslash le\; XV(z)\; \backslash left(\; F\_1\; \backslash left(\backslash frac\; \backslash right)\; +\; O\backslash left(\backslash frac\}\backslash right)\; \backslash right)\; +\; \backslash sum\_\; 4^\; \backslash left|\; r\_A(m)\; \backslash right|$ and : $S(A,P,z)\; \backslash ge\; XV(z)\; \backslash left(\; f\_1\; \backslash left(\backslash frac\; \backslash right)\; -\; O\backslash left(\backslash frac\}\backslash right)\; \backslash right)\; -\; \backslash sum\_\; 4^\; \backslash left|\; r\_A(m)\; \backslash right|.$ 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「Jurkat–Richert 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.