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arXiv · 0905.0318

Mod-Poisson convergence in probability and number theory

E. Kowalski·A. Nikeghbali
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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.

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E. Kowalski, A. Nikeghbali. 2009-12-26. Mod-Poisson convergence in probability and number theory. https://arxiv.org/abs/0905.0318

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