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Mesoscopic fluctuations of the zeta zeros
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  • Published: 07 July 2009

Mesoscopic fluctuations of the zeta zeros

  • P. Bourgade1 

Probability Theory and Related Fields volume 148, pages 479–500 (2010)Cite this article

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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.

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Authors and Affiliations

  1. Telecom ParisTech, 46 rue Barrault, 75634, Paris Cedex 13, France

    P. Bourgade

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  1. P. Bourgade
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Correspondence to P. Bourgade.

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Bourgade, P. Mesoscopic fluctuations of the zeta zeros. Probab. Theory Relat. Fields 148, 479–500 (2010). https://doi.org/10.1007/s00440-009-0237-3

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  • Received: 10 February 2009

  • Revised: 11 June 2009

  • Published: 07 July 2009

  • Issue Date: November 2010

  • DOI: https://doi.org/10.1007/s00440-009-0237-3

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Keywords

  • Central limit theorem
  • Zeta and L-functions

Mathematics Subject Classification (2000)

  • 11M06
  • 60F05
  • 15A52
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