Mean values of the logarithmic derivative of the Riemann zeta-function near the critical line

IF 0.8 3区数学 Q2 MATHEMATICS

Mathematika Pub Date : 2023-03-23 DOI:10.1112/mtk.12194

Fan Ge

引用次数: 3

Abstract

Assuming the Riemann hypothesis and a hypothesis on small gaps between zeta zeros (see equation (ES 2K) below for a precise definition), we prove a conjecture of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith [J. Math. Phys. 60 (2019), no. 8, 083509], which states that for any positive integer K and real number $a > 0$ ,

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临界线附近黎曼ζ函数对数导数的平均值

假设黎曼假设和关于zeta 0之间小间隙的假设(精确定义见下面的方程(ES 2K))，我们证明了Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein和Snaith的一个猜想[J]。数学。物理60 (2019)，no。[8,083509]，表明对于任意正整数K和实数a>0 $a>0$, lima→0+limT→∞(2a)2K−1T(logT)2K∫T2Tζ ' ζ12+alogT+it2Kdt=2K−2K−1。$$\begin{align*} &\lim _{a \rightarrow 0^+}\lim _{T \rightarrow \infty } \frac{(2a)^{2K-1}}{T (\log T)^{2K}} \int _{T}^{2T} {\left|\frac{\zeta ^{\prime }}{\zeta }{\left(\frac{1}{2}+\frac{a}{\log T}+it\right)}\right|}^{2K} dt\\ &\quad = \binom{2K-2}{K-1}. \end{align*}$$当K=1 $K=1$时，这基本上是Goldston, Gonek和Montgomery的结果[J]。莱恩·安格。[数学学报，537(2001)，105-126](见式(1))。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematika MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

1.40

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.