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Strong Szegő Asymptotics and Zeros of the Zeta‐Function
Author(s) -
Bourgade Paul,
Kuan Jeffrey
Publication year - 2014
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21475
Subject(s) - mathematics , toeplitz matrix , riemann zeta function , riemann hypothesis , pure mathematics , gaussian , mathematical analysis , quantum mechanics , physics
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.