Strong Szego asymptotics and zeros of L-functions

Details Strong Szego asymptotics and zeros of L-functions Strong Szego asymptotics and zeros of L-functions

2012-03-23

arxiv.org/abs/1203.5328v2

Mathematics

Probability

Scientific paper

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 $\zeta'/\zeta$ and the Helffer-Sj\"ostrand functional calculus. Our main result is an analogue of the strong Szeg{\H o} theorem, known for Toeplitz operators and random matrix theory.

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