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

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 ${\max }_{|h|\le 1/2}|\zeta (1/2+it+ih)|$, as $T\le t\le 2T$ varies. In particular, we prove that for $(1+o(1))T$ values of $T\le t\le 2T$ we have ${\max }_{|h|\le 1/2}\log |\zeta (1/2+it+ih)|\le \log \log T-(3/4+o(1))\log \log \log T$, 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 $T\le t\le 2T$ (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.

arXiv comment40 pages

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• The Fyodorov-Hiary-Keating Conjecture. I
• Critical Gaussian multiplicative chaos: a review
• Statistics of extremes in eigenvalue-counting staircases
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• Partial sums of random multiplicative functions and extreme values of a model for the Riemann zeta function
• On the moments of the moments of the characteristic polynomials of Haar distributed symplectic and orthogonal matrices
• A model problem for multiplicative chaos in number theory
• Maxima of log-correlated fields: some recent developments
• Universality of the minimum modulus for random trigonometric polynomials
• Evidence of Random Matrix Corrections for the Large Deviations of Selberg's Central Limit Theorem
• On the moments of moments of random matrices and Ehrhart polynomials