Strong Szegő Asymptotics and Zeros of the Zeta‐Function

Assuming the Riemann hypothesis, we prove the weak convergence of linear statistics of the zeros of the Riemann ζ‐function to a Gaussian field, with covariance structure corresponding to the H 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 Szegő theorem, known for Toeplitz operators and random matrix theory. © 2014 Wiley Periodicals, Inc.