On the number variance of zeta zeros and a conjecture of Berry

Meghann Moriah Lugar, Micah B. Milinovich*, Emily Quesada-Herrera

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Assuming the Riemann hypothesis, we prove estimates for the variance of the real and imaginary part of the logarithm of the Riemann zeta function in short intervals. We give three different formulations of these results. Assuming a conjecture of Chan for how often gaps between zeros can be close to a fixed non-zero value, we prove a conjecture of Berry (1988) for the number variance of zeta zeros in the non-universal regime. In this range, Gaussian unitary ensemble statistics do not describe the distribution of the zeros. We also calculate lower order terms in the second moment of the logarithm of the modulus of the Riemann zeta function on the critical line. Assuming Montgomery's pair correlation conjecture, this establishes a special case of a conjecture of Keating and Snaith (2000).

Original languageEnglish
Pages (from-to)303-348
Number of pages46
Issue number2
Publication statusPublished - Apr 2023


  • Riemann zeta-function
  • Riemann hypothesis
  • Random matrix theory

ASJC Scopus subject areas

  • Mathematics(all)

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