Mesoscopic fluctuations of the zeta zeros

Research output: Contribution to journalArticle

Abstract

We prove a multidimensional extension of Selberg's central limit theorem for log ζ, in which non-trivial correlations appear. In particular, this answers a question by Coram and Diaconis about the mesoscopic fluctuations of the zeros of the Riemann zeta function. Similar results are given in the context of random matrices from the unitary group. This shows the correspondence n ↔ log t not only between the dimension of the matrix and the height on the critical line, but also, in a local scale, for small deviations from the critical axis or the unit circle.

Original languageEnglish (US)
Pages (from-to)479-500
Number of pages22
JournalProbability Theory and Related Fields
Volume148
Issue number3-4
DOIs
StatePublished - Nov 2010

Fingerprint

Small Deviations
Unitary group
Unit circle
Random Matrices
Riemann zeta function
Central limit theorem
Correspondence
Fluctuations
Line
Zero
Context
Deviation

Keywords

  • Central limit theorem
  • Zeta and L-functions

ASJC Scopus subject areas

  • Statistics, Probability and Uncertainty
  • Analysis
  • Statistics and Probability

Cite this

Mesoscopic fluctuations of the zeta zeros. / Bourgade, Paul.

In: Probability Theory and Related Fields, Vol. 148, No. 3-4, 11.2010, p. 479-500.

Research output: Contribution to journalArticle

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