Strong Szegő asymptotics and zeros of the zeta function
نویسندگان
چکیده
Assuming the Riemann hypothesis, we prove the weak convergence of linear statistics of the zeros of L-functions to a Gaussian field, with covariance structure corresponding to the Hnorm 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. AMS Subject Classification (2010): 11M06, 11M50, 15B52.
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تاریخ انتشار 2012