The Riemann zeta function and Gaussian multiplicative chaos: statistics on the critical line

TL;DR

This paper establishes a rigorous connection between the Riemann zeta function and Gaussian multiplicative chaos, demonstrating their statistical equivalence on the critical line and mesoscopic scales, linking number theory with probabilistic chaos theory.

Contribution

It proves the convergence of the zeta function to a product involving Gaussian multiplicative chaos and extends this to mesoscopic scales, also relating it to random matrix theory.

Findings

Zeta function converges to a product of a smooth function and a Gaussian chaos distribution.

On mesoscopic scales, zeta behaves like a scalar multiple of Gaussian chaos.

Similar results hold for characteristic polynomials of random unitary matrices.

Abstract

We prove that if $\omega$ is uniformly distributed on $[0,1]$, then as $T\to\infty$, $t\mapsto \zeta(i\omega T+it+1/2)$ converges to a non-trivial random generalized function, which in turn is identified as a product of a very well behaved random smooth function and a random generalized function known as a complex Gaussian multiplicative chaos distribution. This demonstrates a novel rigorous connection between number theory and the theory of multiplicative chaos -- the latter is known to be connected to many other areas of mathematics. We also investigate the statistical behavior of the zeta function on the mesoscopic scale. We prove that if we let $\delta_T$ approach zero slowly enough as $T\to\infty$, then $t\mapsto \zeta(1/2+i\delta_T t+i\omega T)$ is asymptotically a product of a divergent scalar quantity suggested by Selberg's central limit theorem and a strictly Gaussian multiplicative chaos. We also prove a similar result for the characteristic polynomial of a Haar distributed random unitary matrix, where the scalar quantity is slightly different but the multiplicative chaos part is identical. This essentially says that up to scalar multiples, the zeta function and the characteristic polynomial of a Haar distributed random unitary matrix have an identical distribution on the mesoscopic scale.