### 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 language | English (US) |
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Pages (from-to) | 479-500 |

Number of pages | 22 |

Journal | Probability Theory and Related Fields |

Volume | 148 |

Issue number | 3-4 |

DOIs | |

State | Published - Nov 2010 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 148, no. 3-4, pp. 479-500. https://doi.org/10.1007/s00440-009-0237-3

}

TY - JOUR

T1 - Mesoscopic fluctuations of the zeta zeros

AU - Bourgade, Paul

PY - 2010/11

Y1 - 2010/11

N2 - 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.

AB - 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.

KW - Central limit theorem

KW - Zeta and L-functions

UR - http://www.scopus.com/inward/record.url?scp=77955771547&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77955771547&partnerID=8YFLogxK

U2 - 10.1007/s00440-009-0237-3

DO - 10.1007/s00440-009-0237-3

M3 - Article

VL - 148

SP - 479

EP - 500

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 3-4

ER -