On the partition function of the Riemann zeta function, and the Fyodorov--Hiary--Keating conjecture

Abstract

We investigate the "partition function" integrals 1/2 −1/2 |ζ(1/2+it+ih)| 2 dh for the critical exponent 2, and the local maxima max |h|≤1/2 |ζ(1/2 + it + ih)|, as T ≤ t ≤ 2T varies. In particular, we prove that for (1+o(1))T values of T ≤ t ≤ 2T we have max |h|≤1/2 log |ζ(1/2+it+ih)| ≤ log log T −(3/4+o(1)) log log log T , matching for the first time with both the leading and second order terms predicted by a conjecture of Fyodorov, Hiary and Keating.The proofs work by approximating the zeta function in mean square by the product of a Dirichlet polynomial over smooth numbers and one over rough numbers. They then apply ideas and results from corresponding random model problems to compute averages of this product, under size restrictions on the smooth part that hold for most T ≤ t ≤ 2T (but reduce the size of the averages). There are connections with the study of critical multiplicative chaos. Unlike in some previous work, our arguments never shift away from the critical line by more than a tiny amount 1/ log T , and they don't require explicit calculations of Fourier transforms of Dirichlet polynomials.

A model problem for multiplicative chaos in number theory

“…There has been a lot of recent progress towards the conjectures in (1.6) and (1.7) and other related questions, see [1], [2], [3], [4], [15], [22], [29]. Most notably, the upper bound portion of (1.7) has been established by Harper [21], who also established a slightly weaker version of the upper bound in (1.6). An even more precise version of the upper bound in (1.6) has been established by Arguin, Bourgade and Radziwi l l [3].…”

Maxima of log-correlated fields: some recent developments

“…It would be interesting to understand (211) for general k. Indeed, conjecture 2.2 implicitly assumes that k ≥ 1 (but not that k is integral). If the analogy with number theory holds, then one should expect a correction to conjecture 2.2 for k ∈ (0, 1) and β = 1 after the result of Harper [89], cf. theorem 2.11.…”

“…For (265) one has Choosing to integrate over intervals of length 1 in the integrand of (276) may be easily extended to any interval of O(1), with the appropriate normalization. Harper [89] determined an asymptotic upper bound on (276) for a particular parameter range. Harper argues that this upper bound is likely to be best possible.…”

Universality of the minimum modulus for random trigonometric polynomials

“…Sharp asymptotics for the maximum of random trigonometric polynomials with Rademacher coefficients were obtained by Salem and Zygmund [SZ54] and Halász [Hal73], and extended to more general coefficient distributions by Kahane [Kah85]. In recent years there has been particular focus on characteristic polynomials of random unitary matrices, with γ the unit circle [ABB17, PZ17, CMN18, CZ20], and the Riemann zeta function on a randomly shifted unit interval on the critical axis [ABB + 19,Naj18,Har,ABR]. Such questions are closely tied to a fine understanding of large deviations and concentration of measure for values of the function at given points.…”