Séminaire du Centre de Mathématiques de l’Université Cadi Ayyad pour l’année 2022-2023

Titre:  A walk on Legendre paths

Intervenant: Pr. Youness Lamzouri  (Institut Élie Cartan de Lorraine, Nancy, France )

Date et heure: Le vendredi 24/02/2023 de 10h30- 11h30

Lieu: à la salle de séminaire du département de Mathématiques de la FSSM.

 Résumé :  The Legendre symbol is one of the most basic, mysterious and extensively studied objects in number theory. It is a multiplicative function that encodes information about whether an integer is a square modulo an odd prime p. The Legendre symbol was introduced by Adrien-Marie Legendre in 1798, and has since found countless applications in various areas of mathematics as well as in other fields including cryptography. In this talk, we shall explore what we call “Legendre paths”, which encode information about the values of the Legendre symbol. The Legendre path modulo p is defined as the polygonal path in the plane formed by joining the partial sums of the Legendre symbol modulo p. 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? and what proportion of the path is above the x-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. 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 Rademakher random multiplicative functions.