Distribution of short sums of classical Kloosterman sums of prime powers moduli
Annales mathématiques Blaise Pascal, Volume 26 (2019) no. 1, pp. 101-117.

In [13], the author proved, under some very general conditions, that short sums of -adic trace functions over finite fields of varying center converges in law to a Gaussian random variable or vector. The main inputs are P. Deligne’s equidistribution theorem, N. Katz’ works and the results surveyed in [3]. In particular, this applies to 2-dimensional Kloosterman sums Kl 2,𝔽 q studied by N. Katz in [6] and in [7] when the field 𝔽 q gets large.

This article considers the case of short sums of normalized classical Kloosterman sums of prime powers moduli Kl p n , as p tends to infinity among the prime numbers and n2 is a fixed integer. A convergence in law towards a real-valued standard Gaussian random variable is proved under some very natural conditions.

Dans [13], l’auteur démontre, sous des hypothèses très générales, que les sommes courtes des fonctions traces -adiques sur des corps finis de centre variable convergent en loi vers une variable aléatoire gaussienne ou un vecteur aléatoire gaussien. Les ingrédients principaux sont le théorème d’équirépartition de P. Deligne, les travaux de N. Katz et les résultats présentés dans [3]. Ceci s’applique en particulier au sommes de Kloosterman Kl 2,𝔽 q de dimension 2 étudiées par N. Katz dans [6] et [7] lorsque le corps 𝔽 q grandit.

Dans cet article, on considère le cas des sommes courtes des sommes de Kloosterman normalisées de module une puissance d’un nombre premier Kl p n , lorsque p tend vers l’infini parmi les nombres premiers et n2 est un entier fixé. Sous des hypothèses très naturelles, on démontre la convergence en loi vers une variable aléatoire gaussienne réelle standard.

Published online:
DOI: 10.5802/ambp.385
Classification: 11T23,  11L05
Keywords: Kloosterman sums, moments
Guillaume Ricotta 1

1 Université de Bordeaux Institut de Mathématiques de Bordeaux 351, cours de la Libération 33405 Talence cedex FRANCE
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Guillaume Ricotta. Distribution of short sums of classical Kloosterman sums of prime powers moduli. Annales mathématiques Blaise Pascal, Volume 26 (2019) no. 1, pp. 101-117. doi : 10.5802/ambp.385. https://ambp.centre-mersenne.org/articles/10.5802/ambp.385/

[1] Rabi N. Bhattacharya; R. Ranga Rao Normal approximation and asymptotic expansions, Robert E. Krieger Publishing Co., 1986, xiv+291 pages (Reprint of the 1976 original) | MR | Zbl

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

[3] Étienne Fouvry; Emmanuel Kowalski; Philippe Michel A study in sums of products, Philos. Trans. A, R. Soc. Lond., Volume 373 (2015) no. 2040, 20140309, 26 pages | DOI | MR | Zbl

[4] Allan Gut Probability: a graduate course, Springer Texts in Statistics, Springer, 2005, xxiv+603 pages | MR | Zbl

[5] Henryk Iwaniec; Emmanuel Kowalski Analytic number theory, Colloquium Publications, 53, American Mathematical Society, 2004, xii+615 pages | MR | Zbl

[6] Nicholas M. Katz Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies, 116, Princeton University Press, 1988, x+246 pages | DOI | MR | Zbl

[7] Nicholas M. Katz Exponential sums and differential equations, Annals of Mathematics Studies, 124, Princeton University Press, 1990, xii+430 pages | DOI | MR | Zbl

[8] Emmanuel Kowalski Arithmetic randonnée. An introduction to probabilistic number theory (2016) (https://people.math.ethz.ch/~kowalski/probabilistic-number-theory.pdf)

[9] Youness Lamzouri The distribution of short character sums, Math. Proc. Camb. Philos. Soc., Volume 155 (2013) no. 2, pp. 207-218 | DOI | MR | Zbl

[10] Youness Lamzouri Prime number races with three or more competitors, Math. Ann., Volume 356 (2013) no. 3, pp. 1117-1162 | DOI | MR | Zbl

[11] Kit-Ho Mak; Alexandru Zaharescu The distribution of values of short hybrid exponential sums on curves over finite fields, Math. Res. Lett., Volume 18 (2011) no. 1, pp. 155-174 | DOI | MR | Zbl

[12] Philippe Michel Minorations de sommes d’exponentielles, Duke Math. J., Volume 95 (1998) no. 2, pp. 227-240 | DOI | MR | Zbl

[13] Corentin Perret-Gentil Gaussian distribution of short sums of trace functions over finite fields, Math. Proc. Camb. Philos. Soc., Volume 163 (2017) no. 3, pp. 385-422 | DOI | MR | Zbl

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

[15] Guillaume Ricotta; Emmanuel Royer; Igor Shparlinski Kloosterman paths of prime powers moduli, II (https://arxiv.org/abs/1810.01150, to appear in Bull. Soc. Math. Fr.) | Zbl

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