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November 2021 Moments of the Riemann zeta function on short intervals of the critical line
Louis-Pierre Arguin, Frédéric Ouimet, Maksym Radziwiłł
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Ann. Probab. 49(6): 3106-3141 (November 2021). DOI: 10.1214/21-AOP1524

Abstract

We show that as T, for all t[T,2T] outside of a set of measure o(T),

logθTlogθT|ζ(12+it+ih)|βdh=(logT)fθ(β)+o(1),

for some explicit exponent fθ(β), where θ>1 and β>0. This proves an extended version of a conjecture of Fyodorov and Keating (Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014) 20120503, 32). In particular, it shows that, for all θ>1, the moments exhibit a phase transition at a critical exponent βc(θ), below which fθ(β) is quadratic and above which fθ(β) is linear. The form of the exponent fθ also differs between mesoscopic intervals (1<θ<0) and macroscopic intervals (θ>0), a phenomenon that stems from an approximate tree structure for the correlations of zeta. We also prove that, for all t[T,2T] outside a set of measure o(T),

max|h|logθT|ζ(12+it+ih)|=(logT)m(θ)+o(1),

for some explicit m(θ). This generalizes earlier results of Najnudel (Probab. Theory Related Fields 172 (2018) 387–452) and Arguin et al. (Comm. Pure Appl. Math. 72 (2019) 500–535) for θ=0. The proofs are unconditional, except for the upper bounds when θ>3, where the Riemann hypothesis is assumed.

Funding Statement

L.-P. A. is supported in part by NSF Grant DMS-1513441 and by NSF CAREER DMS-1653602. F. O. is supported by postdoctoral fellowships from the NSERC (PDF) and the FRQNT (B3X). M. R. acknowledges support of a Sloan fellowship and NSF Grant DMS-1902063.

Citation

Download Citation

Louis-Pierre Arguin. Frédéric Ouimet. Maksym Radziwiłł. "Moments of the Riemann zeta function on short intervals of the critical line." Ann. Probab. 49 (6) 3106 - 3141, November 2021. https://doi.org/10.1214/21-AOP1524

Information

Received: 1 January 2021; Published: November 2021
First available in Project Euclid: 7 December 2021

Digital Object Identifier: 10.1214/21-AOP1524

Subjects:
Primary: 60G70
Secondary: 11M06 , 60F10 , 60G60

Keywords: Extreme value theory , moments , Riemann zeta function

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 6 • November 2021
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