Mod-Poisson convergence in probability and number theory

2009/05/04 by Emmanuel Kowalski, Kowalski, E., Ashkan Nikeghbali +1 · 2 citations
Mathematics · #11N25 #11T55 #14G10 #60E10 #60F05 #60F15 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.0905.0318

openalex publication_date 2009/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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