Abstract
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.
Original language | English (US) |
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Pages (from-to) | 1028-1044 |
Number of pages | 17 |
Journal | Communications on Pure and Applied Mathematics |
Volume | 67 |
Issue number | 6 |
DOIs | |
State | Published - Jun 2014 |
ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics