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

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

2008-07-29

arxiv.org/abs/0807.4739v2

Mathematics

Number Theory

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

Scientific paper

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 Erd\H{o}s-K\'ac Theorem.

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