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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 De Bruijn–Newman constant ： ウィキペディア英語版 De Bruijn–Newman constant The De Bruijn–Newman constant, denoted by Λ and named after Nicolaas Govert de Bruijn and Charles M. Newman, is a mathematical constant defined via the zeros of a certain function ''H''(''λ'', ''z''), where ''λ'' is a real parameter and ''z'' is a complex variable. ''H'' has only real zeros if and only if ''λ'' ≥ Λ. The constant is closely connected with Riemann's hypothesis concerning the zeros of the Riemann zeta-function. In brief, the Riemann hypothesis is equivalent to the conjecture that Λ ≤ 0. De Bruijn showed in 1950 that ''H'' has only real zeros if ''λ'' ≥ 1/2, and moreover, that if ''H'' has only real zeros for some λ, ''H'' also has only real zeros if λ is replaced by any larger value. Newman proved in 1976 the existence of a constant Λ for which the "if and only if" claim holds; and this then implies that Λ is unique. Newman conjectured that Λ ≥ 0, an intriguing counterpart to the Riemann hypothesis. Serious calculations on lower bounds for Λ have been made since 1988 and—as can be seen from the table—are still being made: Since $H(\backslash lambda\; ,\; z)$ is just the Fourier transform of $F(e^\backslash Phi)$ then ''H'' has the Wiener–Hopf representation: :$\backslash xi\; (1/2+iz)=\; A\backslash sqrt\; \backslash pi\; (\backslash lambda)^\; \backslash int\_^\backslash infty\; e^(x-z)^\}\; H(\backslash lambda\; ,\; x)\; \backslash ,\; dx$ which is only valid for lambda positive or 0, it can be seen that in the limit lambda tends to zero then $H(0,x)=\backslash xi(1/2+ix)$ for the case Lambda is negative then H is defined so: :$H(z,\backslash lambda)=B\backslash sqrt\; \backslash pi\; (\backslash lambda)^\; \backslash int\_^\backslash infty\; e^(x-z)^\}\; \backslash xi(1/2+ix)\; \backslash ,\; dx$ where ''A'' and ''B'' are real constants. ==References== * * * * 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「De Bruijn–Newman constant」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.