THE RIEMANN ZETA FUNCTION AND GAUSSIAN MULTIPLICATIVE CHAOS: STATISTICS ON THE CRITICAL LINE

Abstract

We prove that if ω is uniformly distributed on [0, 1], then as T →∞, t → ζ(iωT + 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.