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On the extreme values of the Riemann zeta function on random intervals of the critical line

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Abstract

In the present paper, we show that under the Riemann hypothesis, and for fixed \(h, \epsilon > 0\), the supremum of the real and the imaginary parts of \(\log \zeta (1/2 + it)\) for \(t \in [UT -h, UT + h]\) are in the interval \([(1-\epsilon ) \log \log T, (1+ \epsilon ) \log \log T]\) with probability tending to 1 when T goes to infinity, U being a uniform random variable in [0, 1]. This proves a weak version of a conjecture by Fyodorov, Hiary and Keating, which has recently been intensively studied in the setting of random matrices. We also unconditionally show that the supremum of \(\mathfrak {R}\log \zeta (1/2 + it)\) is at most \(\log \log T + g(T)\) with probability tending to 1, g being any function tending to infinity at infinity.

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Acknowledgements

The author would like to thank R. Chhaibi for helpful discussions we had on the problem solved in the present article, and the referees for their comments and suggestions, which have greatly improved the writing of this paper. One of the referees suggested some of the questions stated at the end of the introduction.

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  1. Department of Mathematical Science, University of Cincinnati, 2600 Clifton Avenue, Cincinnati, 45221, OH, USA

    Joseph Najnudel

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  1. Joseph Najnudel
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Najnudel, J. On the extreme values of the Riemann zeta function on random intervals of the critical line. Probab. Theory Relat. Fields 172, 387–452 (2018). https://doi.org/10.1007/s00440-017-0812-y

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