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

Eero Saksman*, Christian Webb

*Tämän työn vastaava kirjoittaja

Tutkimustuotos: LehtiartikkeliArticleScientificvertaisarvioitu

Abstrakti

We prove that if omega is uniformly distributed on [0, 1], then as T -> infinity, t bar right arrow zeta (i omega T + 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.

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 -> infinity, then t bar right arrow 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 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.

AlkuperäiskieliEnglanti
Sivut2680-2754
Sivumäärä75
JulkaisuANNALS OF PROBABILITY
Vuosikerta48
Numero6
DOI - pysyväislinkit
TilaJulkaistu - marraskuuta 2020
OKM-julkaisutyyppiA1 Julkaistu artikkeli, soviteltu

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