Strong Szego Asymptotics and Zeros of the Zeta-Function

Paul Bourgade, Jeffrey Kuan

Research output: Contribution to journalArticle

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 languageEnglish (US)
JournalCommunications on Pure and Applied Mathematics
DOIs
StateAccepted/In press - 2013

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Riemann Function
Gaussian Fields
Functional Calculus
Operator Matrix
Riemann hypothesis
Random Matrix Theory
Toeplitz Operator
Toeplitz matrix
Covariance Structure
Test function
Weak Convergence
Riemann zeta function
Explicit Formula
Statistics
Analogue
Norm
Zero
Theorem
Form

ASJC Scopus subject areas

  • Applied Mathematics
  • Mathematics(all)

Cite this

Strong Szego Asymptotics and Zeros of the Zeta-Function. / Bourgade, Paul; Kuan, Jeffrey.

In: Communications on Pure and Applied Mathematics, 2013.

Research output: Contribution to journalArticle

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