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

Abstract

We show that as T→∞, for all t∈[T,2T] outside of a set of measure o(T), ∫−logθTlogθT|ζ(1 2+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(1 2+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.

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