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

Eero Saksman, Christian Webb

Published 2016-08-31Version 1

We establish a functional limit theorem that describes the statistical behavior of the Riemann zeta function on the critical line in a fixed size neighborhood of a generic point. More precisely, our main result shows 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 connection between number theory and the theory of multiplicative chaos -- the latter is known to be connected to many other areas of mathematics. As a by-product of our proof we demonstrate that statistically and in the sense of generalized functions, the truncated Euler products approximate well the Riemann zeta function also on the critical line. We also investigate the statistical behavior of the zeta function on the mesoscopic scale. It will be shown that if we let $\delta_T$ approach zero slowly enough as $T\to\infty$, then $t\mapsto \zeta(1/2+i\delta_T t+i\omega T)$ tends to a product of a scalar quantity suggested by Selberg's central limit theorem and a strictly Gaussian multiplicative chaos appearing on the macroscopic scale. Finally, we conjecture that similar limit theorems are true for objects like $|\zeta|^{\beta}$ for $\beta>0$, and such results could give statistical information about the size of $|\zeta|$ on the critical line. Motivated by this, we construct the conjectured limiting objects and establish their fundamental properties.