Strong Szego asymptotics and zeros of the zeta function

2012/03/23 by Paul Bourgade, Bourgade, Paul, Jeffrey Kuan +1
Mathematics · #Analytic Number Theory Research #Advanced Algebra and Geometry #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1203.5328

Abstract

Assuming the Riemann hypothesis, we prove the weak convergence of linear statistics of the zeros of L-functions towards a Gaussian field, with covariance structure corresponding to the \HH^1/2-norm of the test functions. For this purpose, we obtain an approximate form of the explicit formula, relying on Selberg's smoothed expression for ζ'/ζ and the Helffer-Sjöstrand functional calculus. Our main result is an analogue of the strong Szeg\H o theorem, known for Toeplitz operators and random matrix theory.

Citations

• Toeplitz matrices, translation kernels and a related problem in probability theory
• Hilbert's Inequality

Related

• Selberg's Central Limit Theorem weighted by Linear Statistics of Zeta Zeros
• Linear statistics for zeros of Riemann's zeta function
• Statistics on Riemann zeros
• Explicit zero density estimate for the Riemann zeta-function near the critical line
• Linear statistics of low-lying zeros of L--functions