On Riemann zeroes, Lognormal Multiplicative Chaos, and Selberg Integral

TL;DR

This paper explores deep connections between Riemann zeroes, multiplicative chaos, and Selberg integrals, proposing conjectures and deriving explicit formulas for related statistical measures.

Contribution

It introduces a conjectural link between Riemann zeroes and lognormal multiplicative chaos, and computes key statistical properties using Selberg integral extensions.

Findings

Conjectural relation between Riemann zeroes and multiplicative chaos.

Explicit formulas for moments, covariance, and multiscaling spectrum.

Asymptotic behaviors derived from Selberg integral extensions.

Abstract

Rescaled Mellin-type transforms of the exponential functional of the Bourgade-Kuan-Rodgers statistic of Riemann zeroes are conjecturally related to the distribution of the total mass of the limit lognormal stochastic measure of Mandelbrot-Bacry-Muzy. The conjecture implies that a non-trivial, log-infinitely divisible probability distribution is associated with Riemann zeroes. For application, integral moments, covariance structure, multiscaling spectrum, and asymptotics associated with the exponential functional are computed in closed form using the known meromorphic extension of the Selberg integral.