On the Berry conjecture
type: Item
instance of: scholarly article
title: On the Berry conjecture
zbMATH Open document ID: 0918.11050
author: Akio Fujii
published in: Journal of Mathematics of Kyoto University
publication date: 1999-08-23T00:00:00Z
review text: If \(N(T)\) is the usual counting function for zeros of \(\zeta(s)\), and \(B(x)\) is the inverse function for \({t\over 2\pi}\left(\log{t\over 2\pi}-1\right)+{7\over 8}\), the number variance is defined for \(\alpha\neq 0\) by \[ V(\alpha,T)= T^{-1}\int^T_0\Biggl\{N\Biggl(B\Biggl(x+ {\alpha\over 2}\Biggr)\Biggr)- N\Biggl(B\Biggl(x-{\alpha\over 2}\Biggr)\Biggr)- \alpha\Biggr\}^2 dx. \] \textit{M. V. Berry} [Nonlinearity 1, 399-407 (1988; Zbl 0664.10022)] has given explicit functions \(f(\alpha)\) (the `GUE part') and \(g(\alpha,T,\lambda)\) (the `arithmetic part') and conjectured that \(\widetilde V(\alpha,T,\lambda)=f(\alpha)+g(\alpha,T,\lambda)\), for some value in the range \(1\ll\lambda\ll\log T\), whenever \(\alpha\ll T\log T\). This is a much longer range for \(\alpha\) than has been previously studied. The author gives a thorough investigation of the number variance and its conjectural approximations. In particular he shows that the `simplified number variance' \[ T^{-1}\int^T_0\Biggl\{S\Biggl(t+ {2\pi\alpha \over\log(T/2\pi)}\Biggr)- S(t)\Biggr\}^2 dt \] produces the same main terms as does \(f(\alpha)+g(\alpha,t,\lambda)\), if one assumes the Riemann Hypothesis and Montgomery's pair correlation conjecture. Many other useful results are given.
Mathematics Subject Classification ID: 11M26
zbMATH DE Number: 1140495
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: Berry conjecture
zbMATH Keywords: simplified number variance
zbMATH Keywords: zeros
zbMATH Keywords: number variance
zbMATH Keywords: Riemann hypothesis
zbMATH Keywords: pair correlation conjecture
reviewed by: D. R. Heath-Brown
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518398
OpenAlex ID: W1532547196
Wikidata QID: Q122964625
DOI: 10.1215/KJM/1250518398
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description: scientific article; zbMATH DE number 1140495