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– Kac 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.