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 H1/2-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 Szego theorem, known for Toeplitz operators and random matrix theory.
ASJC Scopus subject areas
- Applied Mathematics