Maxima of a random model of the Riemann zeta function over intervals of varying length
Arguin L-P, Dubach G, Hartung L. Maxima of a random model of the Riemann zeta function over intervals of varying length. arXiv, 2103.04817.
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Author
Arguin, Louis-Pierre;
Dubach, GuillaumeISTA 
;
Hartung, Lisa
;
Hartung, LisaDepartment
Abstract
We consider a model of the Riemann zeta function on the critical axis and study its maximum over intervals of length (log T)θ, where θ is either fixed or tends to zero at a suitable rate.
It is shown that the deterministic level of the maximum interpolates smoothly between the ones
of log-correlated variables and of i.i.d. random variables, exhibiting a smooth transition ‘from
3/4 to 1/4’ in the second order. This provides a natural context where extreme value statistics of
log-correlated variables with time-dependent variance and rate occur. A key ingredient of the
proof is a precise upper tail tightness estimate for the maximum of the model on intervals of
size one, that includes a Gaussian correction. This correction is expected to be present for the
Riemann zeta function and pertains to the question of the correct order of the maximum of
the zeta function in large intervals.
Publishing Year
Date Published
2021-03-08
Journal Title
arXiv
Acknowledgement
The research of L.-P. A. is supported in part by the grant NSF CAREER DMS-1653602. G. D. gratefully acknowledges support from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411. The research of L. H. is supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through Project-ID 233630050 -TRR 146, Project-ID 443891315 within SPP 2265 and Project-ID 446173099.
Article Number
2103.04817
IST-REx-ID
Cite this
Arguin L-P, Dubach G, Hartung L. Maxima of a random model of the Riemann zeta function over intervals of varying length. arXiv. doi:10.48550/arXiv.2103.04817
Arguin, L.-P., Dubach, G., & Hartung, L. (n.d.). Maxima of a random model of the Riemann zeta function over intervals of varying length. arXiv. https://doi.org/10.48550/arXiv.2103.04817
Arguin, Louis-Pierre, Guillaume Dubach, and Lisa Hartung. “Maxima of a Random Model of the Riemann Zeta Function over Intervals of Varying Length.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2103.04817.
L.-P. Arguin, G. Dubach, and L. Hartung, “Maxima of a random model of the Riemann zeta function over intervals of varying length,” arXiv. .
Arguin L-P, Dubach G, Hartung L. Maxima of a random model of the Riemann zeta function over intervals of varying length. arXiv, 2103.04817.
Arguin, Louis-Pierre, et al. “Maxima of a Random Model of the Riemann Zeta Function over Intervals of Varying Length.” ArXiv, 2103.04817, doi:10.48550/arXiv.2103.04817.
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