翻訳と辞書 Words near each other ・ Rieslaner ・ Riesling ・ Riesling Trail ・ Riesman ・ Riesman's sign ・ Riesner ・ Riespach ・ Riess ・ Riess spiral ・ Riessersee ・ Rieste ・ Rieste railway station ・ Riester ・ Riesweiler ・ Riesz ・ Riesz function ・ Riesz mean ・ Riesz potential ・ Riesz rearrangement inequality ・ Riesz representation theorem ・ Riesz sequence ・ Riesz space ・ Riesz theorem ・ Riesz transform ・ Riesz's lemma ・ Riesz–Fischer theorem ・ Riesz–Markov–Kakutani representation theorem ・ Riesz–Thorin theorem ・ Riet ・ Riet River Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Riesz function ： ウィキペディア英語版 Riesz function In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series :$(x)\; =\; -\backslash sum\_^\backslash infty\; \backslash frac.$ If we set $F(x)\; =\; \backslash frac12\; (4\; \backslash pi^2\; x)$ we may define it in terms of the coefficients of the Laurent series development of the hyperbolic (or equivalently, the ordinary) cotangent around zero. If :$\backslash frac\; \backslash coth\; \backslash frac\; =\; \backslash sum\_^\backslash infty\; c\_n\; x^n\; =\; 1\; +\; \backslash frac\; x^2\; -\; \backslash fracx^4\; +\; \backslash cdots$ then ''F'' may be defined as :$F(x)\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; =\; 12x\; -\; 720x^2\; +\; 15120x^3\; -\; \backslash cdots$ The values of ζ(2k) approach one for increasing k, and comparing the series for the Riesz function with that for $x\backslash \; \backslash exp(-x)$ shows that it defines an entire function. Alternatively, ''F'' may be defined as :$F(x)\; =\; \backslash sum\_^\backslash frac\}\}$ denotes the rising factorial power in the notation of D. E. Knuth and the number ''B''''n'' are the Bernoulli number. The series is one of alternating terms and the function quickly tends to minus infinity for increasingly negative values of ''x''. Positive values of ''x'' are more interesting and delicate. ==Riesz criterion== It can be shown that :$\backslash operatorname(x)\; =\; O(x^e)\backslash qquad\; (\backslash textx\backslash to\backslash infty)$ for any exponent ''e'' larger than 1/2, where this is big O notation; taking values both positive and negative. Riesz showed that the Riemann hypothesis is equivalent to the claim that the above is true for any ''e'' larger than 1/4.〔M. Riesz, «Sur l'hypothèse de Riemann», ''Acta Mathematica'', 40 (1916), pp.185-90.». For English translation look (here )〕 In the same paper, he added a slightly pessimistic note too: «''Je ne sais pas encore decider si cette condition facilitera la vérification de l'hypothèse''». 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「Riesz 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.