BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Youness Lamzouri (Institut Élie Cartan de Lorraine (IECL) of the Université de Lorraine in Nancy) DTSTART:20230109T190000Z DTEND:20230109T200000Z DTSTAMP:20260423T021142Z UID:NTC/14 DESCRIPTION:Title: A w alk on Legendre paths\nby Youness Lamzouri (Institut Élie Cartan de L orraine (IECL) of the Université de Lorraine in Nancy) as part of Lethbri dge number theory and combinatorics seminar\n\nLecture held in University of Lethbridge: M1040 (Markin Hall).\n\nAbstract\nThe Legendre symbol is on e of the most basic\, mysterious and extensively studied objects in number theory. It is a multiplicative function that encodes information about wh ether 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 coun tless applications in various areas of mathematics as well as in other fie lds including cryptography. In this talk\, we shall explore what we call ` `Legendre paths''\, which encode information about the values of the Legen dre 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 mod ulo $p$. In particular\, we will attempt to answer the following questions as we vary over the primes $p$: how are these paths distributed? how do t heir maximums behave? and what proportion of the path is above the real ax is? Among our results\, we prove that these paths converge in law\, in the space of continuous functions\, to a certain random Fourier series constr ucted using Rademakher random multiplicative functions. Part of this work is joint with Ayesha Hussain.\n\nThis talk is part of the PIMS Distinguish ed Speaker Series. The registration link is only valid for this talk.\n LOCATION:https://researchseminars.org/talk/NTC/14/ END:VEVENT END:VCALENDAR