Mod-Poisson convergence in probability and number theory

Abstract

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 Erdős-Kác 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.

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

References in corpus (2)

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

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• Mod-Gaussian convergence: new limit theorems in probability and number theory
• Mod-discrete expansions
• The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles
• On closeness of two discrete weighted sums