Probability and Mathematical Physics Seminar

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Large deviation estimates for Selberg’s central limit theorem and applications

Speaker: Emma Bailey, CUNY Graduate Center

Location: Warren Weaver Hall 1302

Date: Friday, November 11, 2022, 11:10 a.m.

Synopsis:

Abstract:

Selberg’s central limit theorem gives that the logarithm of the Riemann zeta function taken at a uniformly drawn height in $[T, 2T]$ behaves as a complex centered Gaussian random variable with variance $\log\log T$.  A natural question is to investigate how far the Gaussian decay persists.  We present results on the right tail for the real part of the logarithm, where the absolute value of zeta is `unusually large’, on the scale of the exponential of the variance. Our proof employs a recursive scheme of Arguin, Bourgade and Radziwi{\l}{\l} to inductively work with the logarithm of zeta, interpreted as a random walk. The result is in agreement with the corresponding (known) random matrix result, under the usual dictionary, and has a number of nice corollaries. This work is joint with Louis-Pierre Arguin.