Mod-Poisson convergence in probability and number theory

Details Mod-Poisson convergence in probability and number theory Mod-Poisson convergence in probability and number theory

2009-05-04

arxiv.org/abs/0905.0318v2

Mathematics

Number Theory

30 pages Version 2 with a few corrections, and added references

Scientific paper

Building on earlier work introducing the notion of "mod-Gaussian" convergence of sequences of random variables, which arises naturally in Random Matrix Theory and number theory, we discuss the analogue notion of "mod-Poisson" convergence. We show in particular how it occurs naturally in analytic number theory in the classical Erd\H{o}s-K\'ac Theorem. In fact, this case reveals deep connections and analogies with conjectures concerning the distribution of L-functions on the critical line, which belong to the mod-Gaussian framework, and with analogues over finite fields, where it can be seen as a zero-dimensional version of the Katz-Sarnak philosophy in the large conductor limit.

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