The Fyodorov–Hiary–Keating conjecture on mesoscopic intervals

Abstract

We derive precise upper bounds for the maximum of the Riemann zeta function on a typical short interval of the critical line. We show that for fixed θ∈(−1,0], large T, and y≥2 satisfying y=O(loglogT/logloglogT), the proportion of points t∈[T,2T] for which max|h|≤logθT|ζ(12+it+ih)|>ey·eS(loglogT)|θ|/2 (logT)(1+θ) (loglogT)3/4 is bounded above by a constant times yexp(−2y−y2/((1+θ)loglogT)), where S=S(t) is a quantity whose value distribution is approximately that of a standard Gaussian. Up to a multiplicative constant, this settles the upper bound of a conjecture of Fyodorov–Hiary–Keating which was only known in the leading order for θ∈(−1,0). Using similar techniques, we also derive upper bounds for the second moment of the zeta function on such intervals. We show that for large T, the proportion of t∈[T,2T] for which 1 logθT∫−logθTlogθT|ζ(12+it+ih)|2dh>AeS2|θ|loglogT (logT)(1+θ) loglogT tends to zero as A→∞, for the same S as above. This proves a weak form of another conjecture of Fyodorov–Keating and generalizes a result of Harper, which is recovered at θ=0 (in which case S is defined to be zero). Our proofs use an adaptation of the recursive scheme introduced by one of the authors, Bourgade and Radziwiłł in (Arguin, Bourgade and Radziwiłł (2020)).