### 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 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 Szego theorem, known for Toeplitz operators and random matrix theory.

Original language | English (US) |
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Journal | Communications on Pure and Applied Mathematics |

DOIs | |

State | Accepted/In press - 2013 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Mathematics(all)

### Cite this

*Communications on Pure and Applied Mathematics*. https://doi.org/10.1002/cpa.21475

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

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Strong Szego Asymptotics and Zeros of the Zeta-Function

AU - Bourgade, Paul

AU - Kuan, Jeffrey

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84880577456&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84880577456&partnerID=8YFLogxK

U2 - 10.1002/cpa.21475

DO - 10.1002/cpa.21475

M3 - Article

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

ER -