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

We show that as T→∞, for all t∈[T,2T] outside of a set measure o(T), ∫−logθTlogθT|ζ(1 2+it+ih)|βdh=(logT)fθ(β)+o(1), some explicit exponent fθ(β), where θ>−1 and β>0. This proves an extended version conjecture Fyodorov Keating (Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014) 20120503, 32). In particular, it shows that, θ>−1, the moments exhibit phase transition at critical βc(θ), below which fθ(β) is quadratic above linear. The form fθ also differs between mesoscopic intervals ( −1<θ<0) macroscopic (θ>0), phenomenon stems from approximate tree structure correlations zeta. prove max|h|≤logθT|ζ(1 2+it+ih)|=(logT)m(θ)+o(1), m(θ). generalizes earlier results Najnudel (Probab. Theory Related Fields 172 (2018) 387–452) Arguin et al. (Comm. Pure Appl. 72 (2019) 500–535) θ=0. proofs are unconditional, except upper bounds when θ>3, Riemann hypothesis assumed.