Abstract
We show that as , for all outside of a set of measure ,
for some explicit exponent , where and . 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 , the moments exhibit a phase transition at a critical exponent , below which is quadratic and above which is linear. The form of the exponent also differs between mesoscopic intervals () and macroscopic intervals (), a phenomenon that stems from an approximate tree structure for the correlations of zeta. We also prove that, for all outside a set of measure ,
for some explicit . 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 . The proofs are unconditional, except for the upper bounds when , 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
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
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