2016/08/31 by Eero Saksman, Christian A. Webb, Saksman, Eero +1 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1609.00027
openalex publication_date 2016/08/31 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
We prove that if \ω is uniformly distributed on [0,1], then as\nT\→\∞, t\↦ \ζ(i\ω T+it+1/2) converges to a non-trivial\nrandom generalized function, which in turn is identified as a product of a very\nwell behaved random smooth function and a random generalized function known as\na complex Gaussian multiplicative chaos distribution. This demonstrates a novel\nrigorous connection between number theory and the theory of multiplicative\nchaos -- the latter is known to be connected to many other areas of\nmathematics.\n We also investigate the statistical behavior of the zeta function on the\nmesoscopic scale. We prove that if we let \δT approach zero slowly\nenough as T\→\∞, then t\↦ \ζ(1/2+i\δT t+i\ω T) is\nasymptotically a product of a divergent scalar quantity suggested by Selberg's\ncentral limit theorem and a strictly Gaussian multiplicative chaos. We also\nprove a similar result for the characteristic polynomial of a Haar distributed\nrandom unitary matrix, where the scalar quantity is slightly different but the\nmultiplicative chaos part is identical. This essentially says that up to scalar\nmultiples, the zeta function and the characteristic polynomial of a Haar\ndistributed random unitary matrix have an identical distribution on the\nmesoscopic scale.\n