A note on the maximum of the Riemann zeta function, and log-correlated random variables

Abstract

In recent work, Fyodorov and Keating conjectured the maximum size of $|\zeta (1/2+it)|$ in a typical interval of length O(1) on the critical line. They did this by modelling the zeta function by the characteristic polynomial of a random matrix; relating the random matrix problem to another problem from statistical mechanics; and applying a heuristic analysis of that problem. In this note we recover a conjecture like that of Fyodorov and Keating, but using a different model for $|\zeta (1/2+it)|$ in terms of a random Euler product. In this case the probabilistic model reduces to studying the supremum of Gaussian random variables with logarithmic correlations, and can be analysed rigorously.

26 pages

References in corpus (2)

• Multivariate normal approximation with Stein's method of exchangeable pairs under a general linearity condition
• Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta-Function

Cited by in corpus (5)

• Moments of the Riemann zeta function on short intervals of the critical line
• Maxima of branching random walks with piecewise constant variance
• Comparison Inequalities for Order Statistics of Gaussian Arrays
• Maxima of log-correlated fields: some recent developments
• Large deviations and continuity estimates for the derivative of a random model of $\log |\zeta |$ on the critical line