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arXiv · 2211.14918

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

Meghann Moriah Lugar·Micah B. Milinovich·Emily Quesada-Herrera
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Abstract

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).

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Meghann Moriah Lugar, Micah B. Milinovich, Emily Quesada-Herrera. 2022-11-27. On the number variance of zeta zeros and a conjecture of Berry. https://arxiv.org/abs/2211.14918

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