Strong szego asymptotics and zeros of the zeta-function

Paul Bourgade, Jeffrey Kuan

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)1028-1044
Number of pages17
JournalCommunications on Pure and Applied Mathematics
Issue number6
StatePublished - Jun 2014

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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