math.PR

Title: 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 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 rigorous connection between number theory and the theory of multiplicative chaos -- the latter is known to be connected to many other areas of mathematics.
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 essentially 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.
 Comments: Supersedes arXiv:1604.08378. Version 2 contains a limit theorem describing the mesoscopic behavior of the characteristic polynomial of random unitary matrices Subjects: Probability (math.PR); Number Theory (math.NT) Cite as: arXiv:1609.00027 [math.PR] (or arXiv:1609.00027v3 [math.PR] for this version)

Submission history

From: Christian Webb [view email]
[v1] Wed, 31 Aug 2016 20:12:20 GMT (67kb)
[v2] Wed, 8 Feb 2017 13:23:18 GMT (75kb)
[v3] Thu, 22 Feb 2018 11:14:05 GMT (76kb)

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