Selberg's Central Limit Theorem weighted by Linear Statistics of Zeta Zeros

2025/07/05 by Fazzari, Alessandro, Gerspach, Maxim, Minelli, Paolo
#FOS: Mathematics #Number Theory (math.NT) #Primary 11M06 #Secondary 11M26

paper · doi:10.48550/arxiv.2507.04150

Abstract

We consider the value distribution of the logarithm of the Riemann zeta function on the critical line, weighted by the local statistics of zeta zeros. We show that, with appropriate normalization, it satisfies a complex Central Limit Theorem, provided that the Fourier support of the test function in the linear statistics is sufficiently small. For the imaginary part, we extend this support condition up to its natural barrier under the Riemann Hypothesis. Finally, we prove that the correlation between log ζ and the one-level density, while negligible on the level of Selberg's Central Limit Theorem, only decays at a rather slow rate if the Riemann Hypothesis is assumed. Our results can be viewed as a combination of Selberg's Central Limit Theorem with work of Hughes and Rudnick on mock-Gaussian behavior of the local statistics.

Citations

• The third moment of the logarithm of zeta and a twisted pair correlation conjecture
• Lower Bounds for the Large Deviations of Selberg's Central Limit Theorem
• On the joint second moment of zeta and its logarithmic derivative
• On the number variance of zeta zeros and a conjecture of Berry
• A weighted one-level density of the non-trivial zeros of the Riemann zeta-function
• Large Deviation Estimates of Selberg's Central Limit Theorem and Applications
• Weighted central limit theorems for central values of L-functions
• Weighted value distributions of the Riemann zeta function on the\n critical line
• Low-lying zeros of symmetric power L-functions weighted by symmetric square L-values
• Large deviations in Selberg's central limit theorem
• Linear statistics of low-lying zeros of L--functions
• Linear Statistics of Low-Lying Zeros of L-Functions
• Linear statistics for zeros of Riemann's zeta function
• Zeros of principal L-functions and random matrix theory

Related

• Mesoscopic fluctuations of the zeta zeros
• Selberg's central limit theorem for log |ζ(\tfrac 12+it)|
• An asymptotic expansion of Selberg's central limit theorem near the\n critical line
• The a-values of the Riemann zeta function near the critical line
• Explicit zero density estimate for the Riemann zeta-function near the critical line