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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 Hardy–Littlewood zeta-function conjectures ： ウィキペディア英語版 Hardy–Littlewood zeta-function conjectures In mathematics, the Hardy–Littlewood zeta-function conjectures, named after Godfrey Harold Hardy and John Edensor Littlewood, are two conjectures concerning the distances between zeros and the density of zeros of the Riemann zeta function. In 1914 Godfrey Harold Hardy proved that the Riemann zeta function $\backslash zeta\backslash bigl(\backslash tfrac+it\backslash bigr)$ has infinitely many real zeros. Let $N(T)$ be the total number of real zeros, $N\_0(T)$ be the total number of zeros of odd order of the function $\backslash zeta\backslash bigl(\backslash tfrac+it\backslash bigr)$, lying on the interval $(0,T]$. Hardy and Littlewood claimed two conjectures. These conjectures – on the distance between real zeros of $\backslash zeta\backslash bigl(\backslash tfrac+it\backslash bigr)$ and on the density of zeros of $\backslash zeta\backslash bigl(\backslash tfrac+it\backslash bigr)$ on intervals $(T,T+H]$ for sufficiently great $T\; >\; 0$, $H\; =\; T^$ and with as less as possible value of $a\; >\; 0$, where $\backslash varepsilon\; >\; 0$ is an arbitrarily small number – open two new directions in the investigation of the Riemann zeta function. 1. For any $\backslash varepsilon\; >\; 0$ there exists such $T\_0\; =\; T\_0(\backslash varepsilon)\; >\; 0$ that for $T\; \backslash geq\; T\_0$ and $H=T^$ the interval $(T,T+H]$ contains a zero of odd order of the function $\backslash zeta\backslash bigl(\backslash tfrac+it\backslash bigr)$. 2. For any $\backslash varepsilon\; >\; 0$ there exist $T\_0\; =\; T\_0(\backslash varepsilon)\; >\; 0$ and $c\; =\; c(\backslash varepsilon)\; >\; 0$, such that for $T\; \backslash geq\; T\_0$ and $H=T^$ the inequality $N\_0(T+H)-N\_0(T)\; \backslash geq\; cH$ is true. In 1942 Atle Selberg studied the problem 2 and proved that for any $\backslash varepsilon\; >\; 0$ there exists such $T\_0\; =\; T\_0(\backslash varepsilon)\; >\; 0$ and $c\; =\; c(\backslash varepsilon)\; >\; 0$, such that for $T\; \backslash geq\; T\_0$ and $H=T^$ the inequality $N(T+H)-N(T)\; \backslash geq\; cH\backslash log\; T$ is true. In his turn, Selberg claim his conjecture that it's possible to decrease the value of the exponent $a\; =\; 0.5$ for $H=T^$ which was proved forty-two years later by A.A. Karatsuba. ==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「Hardy–Littlewood zeta-function conjectures」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.