Journal Article

Mod-Poisson Convergence in Probability and Number Theory

Emmanuel Kowalski and Ashkan Nikeghbali

in International Mathematics Research Notices

Volume 2010, issue 18, pages 3549-3587
Published in print January 2010 | ISSN: 1073-7928
Published online February 2010 | e-ISSN: 1687-0247 | DOI: http://dx.doi.org/10.1093/imrn/rnq019
Mod-Poisson Convergence in Probability and Number Theory

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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.

Journal Article.  7684 words.  Illustrated.

Subjects: Mathematics

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