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Étienne Fouvry; Emmanuel Kowalski; Philippe Michel; Chandra Sekhar Raju; Joël Rivat; Kannan Soundararajan
On short sums of trace functions
(Sur les sommes courtes de fonctions trace)
Annales de l'institut Fourier, 67 no. 1 (2017), p. 423-449, doi: 10.5802/aif.3087
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Class. Math.: 11L07, 11L05, 11T23
Mots clés: Somme exponentielle courte, fonction trace, lemme de van der Corput, méthode de complétion, hypothèse de Riemann sur les corps finis

Résumé - Abstract

Nous considérons des sommes de fonctions oscillantes sur des intervalles contenus dans un groupe fini cyclique, de taille proche de la racine carrée du cardinal du groupe. Nous démontrons tout d’abord des bornes non-triviales pour tout intervalle de longueur à peine plus grande que cette racine carrée (améliorant l’inégalité de Polyá-Vinogradov) pour les fonctions bornées dont la transformée de Fourier est bornée. Nous démontrons ensuite que l’existence d’une borne non-triviale pour un intervalle de taille un peu plus petite que la racine carrée est une propriété stable par transformation de Fourier. Nous donnons des applications liées aux fonctions trace sur les corps finis.

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