Extreme Value Statistics in Probabilistic Number Theory

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Extreme value statistics is the study of rare events that occur in natural and social systems. The appearance of such rare events may be understood by investigating specific mechanisms arising from the interaction between the components constituting the system. It turns out that a large class of functions in number theory are formidable models to extend the current theory of extreme value statistics. The aim of this project is to develop new probabilistic techniques to describe at various scales the fine statistics of extrema of number-theoretic functions. These techniques will then be applied to related problems in physics and mathematics. The project will provide research training opportunities for undergraduate students and graduate students in computer programming, data science and mathematics.More precisely, the Riemann zeta function (and to a larger extent the Dirichlet L-functions) is a fundamental function in number theory as it encodes the distribution of prime numbers among the integers. Large values of such functions in particular contain information about the location of the primes, yet they remain poorly understood. This research project builds on recent progress to describe the large values of the zeta function in short intervals. The current project aims at developing new probabilistic techniques to describe the precise statistics of the large values of the zeta function as the size of the observation interval is varied. In particular, the research carried out in this project is expected to reveal new hybrid statistics at small scales. The project also seeks to extend the techniques to Dirichlet L-functions, where few results are currently available. The ultimate goal is to build better tools of extreme value theory. The interdisciplinary nature of the project is expected to produce new theoretical and numerical tools in probability as well as in analytic number theory. The theoretical results and numerical experiments in combination will clarify the elusive connection between random matrices and L-functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.