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

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 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 .M S C 2 0 2 0 11M06, 11M26 (primary) INTRODUCTIONUnderstanding the distribution of the zeros of the Riemann zeta function, 𝜁(𝑠), is an important problem in number theory. Let 𝑁(𝑡) be the number of zeros 𝜌 = 𝛽 + 𝑖𝛾 of 𝜁(𝑠) such that 0 < 𝛾 ⩽ 𝑡 and 0 < 𝛽 < 1 (counted with multiplicity, where the zeros with 𝛾 = 𝑡 are counted with weight 1 2 ).