In this talk, we shall explore certain polygonal paths, that we call ”Legendre paths”, which encode important information about the values of the Legendre symbol. More precisely, the Legendre path modulo a prime number p is defined as the polygonal path in the plane whose vertices are the points (j, Sp(j)) for 0≤j≤p-1, where Sp(j) is the (normalized) sum of Legendre symbols (n/p) for n up to j. In particular, we will attempt to answer the following questions as we vary over the primes p: how are these paths distributed? how do their maximums behave? when does a Legendre path decreases for the first time? what is the typical number of x-intercepts of such paths? and what proportion of a Legendre path is above the real axis? We will see that some of these questions correspond to important and longstanding problems in analytic number theory, including understanding the size of the least quadratic non-residue, and improving the Pólya-Vinogradov inequality for character sums. Among our results, we prove that as we average over the primes, the Legendre paths converge in law, in the space of continuous functions, to a certain random Fourier series constructed using Rademacher random multiplicative functions. Part of this work is joint with Ayesha Hussain and with Oleksiy Klurman and Marc Munsch.
This video is part of the Number Theory Web Seminar series.
