Mod-Gaussian convergence: new limit theorems in probability and number theory

Jean Jacod 1 , Emmanuel Kowalski 2  and Ashkan Nikeghbali 3
  • 1 Institut de mathématiques de Jussieu, Université Pierre et Marie Curie, et C.N.R.S. UMR 7586, 175, rue du Chevaleret, 75013 Paris, France.
  • 2 Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland.
  • 3 Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland.

Abstract

We introduce a new type of convergence in probability theory, which we call “mod-Gaussian convergence”. It is directly inspired by theorems and conjectures, in random matrix theory and number theory, concerning moments of values of characteristic polynomials or zeta functions. We study this type of convergence in detail in the framework of infinitely divisible distributions, and exhibit some unconditional occurrences in number theory, in particular for families of L-functions over function fields in the Katz–Sarnak framework. A similar phenomenon of “mod-Poisson convergence” turns out to also appear in the classical Erdős–Kac Theorem.

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Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.

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