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.
CITATION STYLE
Saksman, E., & Webb, C. (2020). THE RIEMANN ZETA FUNCTION AND GAUSSIAN MULTIPLICATIVE CHAOS: STATISTICS ON THE CRITICAL LINE. Annals of Probability, 48(6), 2680–2754. https://doi.org/10.1214/20-AOP1433
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