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), ∫^((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 (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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