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

TL;DR

This paper proves the Fyodorov--Hiary--Keating conjecture's second order term for the maximum of the Riemann zeta function on the critical line, using approximation by Dirichlet polynomials and random models.

Contribution

It provides the first rigorous proof matching the conjectured second order term for the zeta function's maximum, employing novel approximation and probabilistic techniques.

Findings

Maximum of log|ζ(1/2+it+ih)| matches conjectured second order term

Approximation of zeta by Dirichlet polynomials over smooth and rough numbers

Connections established with critical multiplicative chaos

Abstract

We investigate the ``partition function'' integrals $\int_{-1/2}^{1/2} |\zeta(1/2 + it + ih)|^2 dh$ for the critical exponent 2, and the local maxima $\max_{|h| \leq 1/2} |\zeta(1/2 + it + ih)|$, as $T \leq t \leq 2T$ varies. In particular, we prove that for $(1+o(1))T$ values of $T \leq t \leq 2T$ we have $\max_{|h| \leq 1/2} \log|\zeta(1/2+it+ih)| \leq \log\log T - (3/4 + o(1))\log\log\log T$, 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 $T \leq t \leq 2T$ (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.