The limiting distribution of Legendre paths

Ayesha Hussain; Youness Lamzouri

doi:10.5802/jep.260

The limiting distribution of Legendre paths
[La répartition limite des chemins de Legendre]

Ayesha Hussain ¹ ; Youness Lamzouri ²

¹ Department of Mathematics, University of Exeter North Park Road, Exeter, EX4 4QF, U.K.
² Université de Lorraine, CNRS, IECL, and Institut Universitaire de France F-54000 Nancy, France

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 589-611.

Résumé
Abstract

Soient $p$ un nombre premier et $(\frac{\cdot}{p})$ le symbole de Legendre modulo $p$ . Le chemin de Legendre attaché à $p$ est le chemin polygonal dont les sommets sont les sommes de caractères normalisées $\frac{1}{\sqrt{p}} \sum_{n \leq j} (\frac{n}{p})$ pour $0 \leq j \leq p - 1$ . Dans cet article, nous étudions la répartition des chemins de Legendre lorsqu’on varie le premier $p$ dans un intervalle $[Q, 2 Q]$ , où $Q$ est grand. Notre résultat principal montre que lorsque $Q \to \infty$ , ces chemins convergent en loi, dans l’espace des fonctions continues à valeurs réelles sur $[0, 1]$ , vers une certaine série de Fourier aléatoire construite en utilisant des fonctions aléatoires complètement multiplicatives de Rademacher. Ceci a été démontré précédemment par le premier auteur sous l’hypothèse de Riemann généralisée.

Let $p$ be a prime number and $(\frac{\cdot}{p})$ be the Legendre symbol modulo $p$ . The Legendre path attached to $p$ is the polygonal path whose vertices are the normalized character sums $\frac{1}{\sqrt{p}} \sum_{n \leq j} (\frac{n}{p})$ for $0 \leq j \leq p - 1$ . In this paper, we investigate the distribution of Legendre paths as we vary over the primes $p$ such that $Q \leq p \leq 2 Q$ , when $Q$ is large. Our main result shows that as $Q \to \infty$ , these paths converge in law, in the space of real-valued continuous functions on $[0, 1]$ , to a certain random Fourier series constructed using Rademacher random completely multiplicative functions. This was previously proved by the first author under the assumption of the Generalized Riemann Hypothesis.

Reçu le : 2023-04-26
Accepté le : 2024-05-02
Publié le : 2024-05-22

DOI : 10.5802/jep.260

Classification : 11L40, 11N64, 11K65
Keywords: Legendre symbol, character sums, Rademacher random multiplicative functions, random Fourier series
Mot clés : Symbole de Legendre, sommes de caractères, fonctions multiplicatives aléatoires de Rademacher, séries de Fourier aléatoires

Affiliations des auteurs :

Ayesha Hussain ¹ ; Youness Lamzouri ²

¹ Department of Mathematics, University of Exeter North Park Road, Exeter, EX4 4QF, U.K.
² Université de Lorraine, CNRS, IECL, and Institut Universitaire de France F-54000 Nancy, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Ayesha Hussain and Youness Lamzouri},
     title = {The limiting distribution of {Legendre} paths},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Ayesha Hussain; Youness Lamzouri. The limiting distribution of Legendre paths. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 589-611. doi : 10.5802/jep.260. https://jep.centre-mersenne.org/articles/10.5802/jep.260/

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