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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


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


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),


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.


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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.


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

Digital Object Identifier: 10.1214/21-AOP1524

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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