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Forum Mathematicum

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Volume 23, Issue 4


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

Jean Jacod
  • Institut de mathématiques de Jussieu, Université Pierre et Marie Curie, et C.N.R.S. UMR 7586, 175, rue du Chevaleret, 75013 Paris, France.
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/ Emmanuel Kowalski / Ashkan Nikeghbali
Published Online: 2010-04-14 | DOI: https://doi.org/10.1515/form.2011.030


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.

Keywords.: Limit theorems; random matrices; characteristic polynomial; infinitely divisible distributions; zeta and L-functions; Katz–Sarnak philosophy; Erdős–Kac Theorem

About the article

Received: 2008-07-30

Revised: 2009-11-02

Published Online: 2010-04-14

Published in Print: 2011-07-01

Citation Information: Forum Mathematicum, Volume 23, Issue 4, Pages 835–873, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/form.2011.030.

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