We show that as T→∞, for all t∈[T,2T] outside of a set of
measure o(T), ∫−(logT)θ(logT)θ∣ζ(21+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 (2014). 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), ∣h∣≤(logT)θmax∣ζ(21+it+ih)∣=(logT)m(θ)+o(1), for some explicit m(θ). This
generalizes earlier results of Najnudel (2018) and Arguin et al. (2018) for
θ=0. The proofs are unconditional, except for the upper bounds when
θ>3, where the Riemann hypothesis is assumed.Comment: 33 pages, 1 figure; Changes : Minor corrections. The previous
discretization via Sobolev is replaced by a new (and more effective)
discretization procedure using Fourier transform