The limiting distribution of Legendre paths

Hussain, Ayesha; Lamzouri, Youness

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

Let $p$ be a prime number and $\left(\frac{·}{p}\right)$ be the Legendre symbol modulo $p$. The Legendre path attached to $p$ is the polygonal path whose vertices are the normalized character sums $\frac{1}{\sqrt{p}}{\sum }_{n\le j}\left(\frac{n}{p}\right)$ for $0\le j\le p-1$. In this paper, we investigate the distribution of Legendre paths as we vary over the primes $p$ such that $Q\le p\le 2Q$, 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.

Soient $p$ un nombre premier et $\left(\frac{·}{p}\right)$ 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 $\frac{1}{\sqrt{p}}{\sum }_{n\le j}\left(\frac{n}{p}\right)$ pour $0\le j\le p-1$. Dans cet article, nous étudions la répartition des chemins de Legendre lorsqu’on varie le premier $p$ dans un intervalle $[Q,2Q]$, 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.

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

[1] Ankeny, N. C. The least quadratic non residue, Ann. of Math. (2), Volume 55 (1952), pp. 65-72 | DOI | MR | Zbl

[2] Baker, R. C.; Montgomery, Hugh L. Oscillations of quadratic $L$-functions, Analytic number theory (Allerton Park, IL, 1989) (Progress in Math.), Volume 85, Birkhäuser Boston, Boston, MA, 1990, pp. 23-40 | DOI | MR | Zbl

[3] Billingsley, Patrick Convergence of probability measures, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., New York, 1999 | DOI | MR

[4] Bober, Jonathan; Goldmakher, Leo; Granville, Andrew; Koukoulopoulos, Dimitris The frequency and the structure of large character sums, J. Eur. Math. Soc. (JEMS), Volume 20 (2018) no. 7, pp. 1759-1818 | DOI | MR | Zbl

[5] Burgess, D. A. On character sums and $L$-series. II, Proc. London Math. Soc. (3), Volume 13 (1963), pp. 524-536 | DOI | Zbl | MR

[6] Davenport, Harold Multiplicative number theory, Graduate Texts in Math., 74, Springer-Verlag, New York, 2000 | MR

[7] Davenport, Harold; Erdős, P. The distribution of quadratic and higher residues, Publ. Math. Debrecen, Volume 2 (1952), pp. 252-265 | DOI | MR | Zbl

[8] Graham, S. W.; Ringrose, C. J. Lower bounds for least quadratic nonresidues, Analytic number theory (Allerton Park, IL, 1989) (Progress in Math.), Volume 85, Birkhäuser Boston, Boston, MA, 1990, pp. 269-309 | DOI | MR | Zbl

[9] Granville, Andrew; Mangerel, Alexander P. Three conjectures about character sums, Math. Z., Volume 305 (2023) no. 3, 49, 34 pages | DOI | MR | Zbl

[10] Harper, Adam J. Moments of random multiplicative functions, II: High moments, Algebra Number Theory, Volume 13 (2019) no. 10, pp. 2277-2321 | MR | Zbl | DOI

[11] Harper, Adam J. Moments of random multiplicative functions, I: Low moments, better than squareroot cancellation, and critical multiplicative chaos, Forum Math. Pi, Volume 8 (2020), e1, 95 pages | DOI | Zbl | MR

[12] Harper, Adam J. A note on character sums over short moving intervals, 2022 (30 p.) | arXiv

[13] Heath-Brown, D. R. A mean value estimate for real character sums, Acta Arith., Volume 72 (1995) no. 3, pp. 235-275 | DOI | MR | Zbl

[14] Hussain, Ayesha The limiting distribution of character sums, Internat. Math. Res. Notices (2022) no. 20, pp. 16292-16326 | DOI | MR | Zbl

[15] Kalmynin, A. Long nonnegative sums of Legendre symbols, 2019 (23 p.) | arXiv

[16] Kowalski, Emmanuel An introduction to probabilistic number theory, Cambridge Studies in Advanced Math., 192, Cambridge University Press, Cambridge, 2021 | DOI | MR

[17] Kowalski, Emmanuel; Sawin, William F. Kloosterman paths and the shape of exponential sums, Compositio Math., Volume 152 (2016) no. 7, pp. 1489-1516 | DOI | Zbl | MR

[18] Lamzouri, Youness Extreme values of $\arg L(1,\chi )$, Acta Arith., Volume 146 (2011) no. 4, pp. 335-354 | DOI | MR | Zbl

[19] Lamzouri, Youness The distribution of large quadratic character sums and applications, 2022 (to appear in Algebra Number Theory) | arXiv

[20] Lehmer, D. H. Incomplete Gauss sums, Mathematika, Volume 23 (1976) no. 2, pp. 125-135 | DOI | MR | Zbl

[21] Loxton, J. H. The graphs of exponential sums, Mathematika, Volume 30 (1983) no. 2, pp. 153-163 | DOI | MR | Zbl

[22] Loxton, J. H. The distribution of exponential sums, Mathematika, Volume 32 (1985) no. 1, pp. 16-25 | DOI | MR | Zbl

[23] Milićević, Djordje; Zhang, Sichen Distribution of Kloosterman paths to high prime power moduli, Trans. Amer. Math. Soc. Ser. B, Volume 10 (2023), pp. 636-669 | DOI | Zbl | MR

[24] Montgomery, Hugh L. Topics in multiplicative number theory, Lect. Notes in Math., 227, Springer-Verlag, Berlin-New York, 1971 | MR

[25] Montgomery, Hugh L.; Vaughan, Robert C. Mean values of character sums, Canad. J. Math., Volume 31 (1979) no. 3, pp. 476-487 | DOI | MR | Zbl

[26] Montgomery, Hugh L.; Vaughan, Robert C. Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Math., 97, Cambridge University Press, Cambridge, 2007 | MR

[27] Ricotta, Guillaume; Royer, Emmanuel Kloosterman paths of prime powers moduli, Comment. Math. Helv., Volume 93 (2018) no. 3, pp. 493-532 | DOI | Zbl | MR

[28] Ricotta, Guillaume; Royer, Emmanuel; Shparlinski, Igor Kloosterman paths of prime powers moduli, II, Bull. Soc. math. France, Volume 148 (2020) no. 1, pp. 173-188 | DOI | MR | Zbl

[29] Schmüdgen, Konrad The moment problem, Graduate Texts in Math., 277, Springer, Cham, 2017 | DOI | MR