Statistical Questions in Number Theory and Arithmetic Geometry

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Number theory is a thriving area of mathematics research with deep connections to both computer science and digital security. The principal goal of this project is to deepen our understanding of certain ubiquitous number-theoretic objects (such as elliptic curves, power residues modulo prime numbers, and classical arithmetic functions such as Euler's phi-function), by studying them through a quantitative lens. One novel feature of this work is an approach to these problems that takes advantage of our understanding of the "anatomy of integers" describing the typical way a number breaks down into component parts (prime factors).Three main topics of investigation will be pursued: (1) A detailed study of the value-distribution of arithmetic functions, one particular problem being to pin down precise conditions under which a number has many factorizations. Here "factorizations" may be ordered or unordered, and we may wish to impose certain additional conditions on the parts. (2) The distribution of power residues. (3) The statistical distribution of torsion structures of elliptic curves. For (1) and (3), results from the "anatomy integers" will play a critical role. For (2), the PI plans to supplement the existing approaches to such problems - which have been mostly analytic, involving character sums - with algebraic approaches that make use of higher reciprocity laws.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.