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Mod-poisson convergence in probability and number theory


Kowalski, E; Nikeghbali, A (2010). Mod-poisson convergence in probability and number theory. International Mathematics Research Notices, 2010(18):3549-3587.

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.

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.

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11 citations in Web of Science®
13 citations in Scopus®
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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2010
Deposited On:02 May 2011 14:30
Last Modified:31 Dec 2017 06:28
Publisher:Oxford University Press
ISSN:1073-7928
Publisher DOI:https://doi.org/10.1093/imrn/rnq019

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