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

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-Kác Theorem.

33 pages; to appear in Forum Math This version contains a few minor additions and corrections

References in corpus (3)

• Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field
• Mod-Poisson convergence in probability and number theory
• The fluctuations in the number of points on a hyperelliptic curve over a finite field

Cited by in corpus (4)

• Mod-Poisson convergence in probability and number theory
• Mod-discrete expansions
• On Stein's method and mod-* convergence
• The order of large random permutations with cycle weights