Strong Szego asymptotics and zeros of the zeta function

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.