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

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 nontrivial 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 probabilistic number theory and the theory of multiplicative chaos—the latter is known to be connected to various branches of modern probability theory and mathematical physics.

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 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.