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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 Selberg sieve ： ウィキペディア英語版 Selberg sieve In mathematics, in the field of number theory, the Selberg sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Atle Selberg in the 1940s. ==Description== In terms of sieve theory the Selberg sieve is of ''combinatorial type'': that is, derives from a careful use of the inclusion-exclusion principle. Selberg replaced the values of the Möbius function which arise in this by a system of weights which are then optimised to fit the given problem. The result gives an ''upper bound'' for the size of the sifted set. Let ''A'' be a set of positive integers ≤ ''x'' and let ''P'' be a set of primes. For each ''p'' in ''P'', let ''A''''p'' denote the set of elements of ''A'' divisible by ''p'' and extend this to let ''A''''d'' the intersection of the ''A''''p'' for ''p'' dividing ''d'', when ''d'' is a product of distinct primes from ''P''. Further let A1 denote ''A'' itself. Let ''z'' be a positive real number and ''P''(''z'') denote the product of the primes in ''P'' which are ≤ ''z''. The object of the sieve is to estimate :$S(A,P,z)\; =\; \backslash left\backslash vert\; A\; \backslash setminus\; \backslash bigcup\_\; A\_p\; \backslash right\backslash vert\; .$ We assume that |''A''''d''| may be estimated by :$\backslash left\backslash vert\; A\_d\; \backslash right\backslash vert\; =\; \backslash frac\; X\; +\; R\_d\; .$ where ''f'' is a multiplicative function and ''X''   =   |''A''|. Let the function ''g'' be obtained from ''f'' by Möbius inversion, that is :$g(n)\; =\; \backslash sum\_\; \backslash mu(d)\; f(n/d)$ :$f(n)\; =\; \backslash sum\_\; g(d)$ where μ is the Möbius function. Put :$V(z)\; =\; \backslash sum\_\}\; \backslash frac\; .$ Then :$S(A,P,z)\; \backslash le\; \backslash frac\; +\; O\backslash left(\}\; \backslash left\backslash vert\; R\_\; \backslash right\backslash vert\}\; \backslash right)\; .$ It is often useful to estimate ''V''(''z'') by the bound :$V(z)\; \backslash ge\; \backslash sum\_\; \backslash frac\; .\; \backslash ,$ 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「Selberg sieve」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.