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Moments of the Riemann zeta function on short intervals of the critical line

Arguin, Louis-Pierre and Ouimet, Frédéric and Radziwiłł, Maksym (2019) Moments of the Riemann zeta function on short intervals of the critical line. . (Unpublished)

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We show that as T→∞, for all t∈[T,2T] outside of a set of measure o(T), ∫^((log T)^θ⁰)_(−(log T)^θ) |ζ(1/2 + it + ih)|^β dh =(log T)^(f_θ(β) + o(1)), for some explicit exponent f_θ(β), where θ > −1 and β > 0. This proves an extended version of a conjecture of Fyodorov and Keating (2014). In particular, it shows that, for all θ > −1, the moments exhibit a phase transition at a critical exponent β_c(θ), below which f_θ(β) is quadratic and above which f_θ(β) is linear. The form of the exponent f_θ 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| ≤ (log T)θ) |ζ(1/2 + it + ih)| = (log T)^(m(θ) + o(1)), for some explicit m(θ). This generalizes earlier results of Najnudel (2018) and Arguin et al. (2019) for θ = 0. The proofs are unconditional, except for the upper bounds when θ > 3, where the Riemann hypothesis is assumed.

Item Type:Report or Paper (Discussion Paper)
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URLURL TypeDescription Paper
Ouimet, Frédéric0000-0001-7933-5265
Additional Information:L.-P. A. is supported in part by NSF Grant DMS-1513441 and by NSF CAREER DMS-1653602. F. O. is supported by postdoctoral fellowship from the NSERC (PDF) and the FRQNT (B3X). M. R. acknowledges support of a Sloan fellowship and NSF grant DMS-1902063.
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Fonds de recherche du Québec - Nature et technologies (FRQNT)B3X
Alfred P. Sloan FoundationUNSPECIFIED
Subject Keywords:extreme value theory, Riemann zeta function, moments
Classification Code:MSC 2020 subject classifications: Primary 60G70; Secondary 11M06, 60F10, 60G60.
Record Number:CaltechAUTHORS:20210825-184519912
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110520
Deposited By: George Porter
Deposited On:26 Aug 2021 21:52
Last Modified:26 Aug 2021 21:52

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