Showing a limited preview of this publication:
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.
Keywords.: Limit theorems; random matrices; characteristic polynomial; infinitely divisible distributions; zeta and L-functions; Katz–Sarnak philosophy; Erdős–Kac Theorem
Received: 2008-07-30
Revised: 2009-11-02
Published Online: 2010-04-14
Published in Print: 2011-July
© de Gruyter 2011