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

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

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 nonzero value, we prove a conjecture of Berry (1988) for the number variance of zeta zeros in the non-universal regime. In this range, GUE 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
Number of pages38
Publication statusAccepted/In press - Nov 2022


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


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