On Short Sums Involving Fourier Coefficients of Maass Forms

Jesse Jääsaari

doi:10.5802/jtnb.1142

Jesse Jääsaari ¹

¹ Department of Mathematics and Statistics FI-20014 University of Turku, Finland

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 3, pp. 761-793.

Résumé
Abstract

Nous étudions les sommes des valeurs propres des opérateurs de Hecke des formes paraboliques de Hecke–Maass pour le groupe $SL (n, ℤ)$ avec $n \geq 3$ quelconque, sur des intervalles courts d’une certaine longueur, en admettant l’hypothèse de Lindelöf généralisée et une estimation pour l’exposant en direction de la conjecture de Ramanujan–Petersson, un peu plus forte que celle qui est actuellement connue. En particulier, dans cette situation, nous donnons une évaluation asymptotique du deuxième moment des sommes en question.

We study sums of Hecke eigenvalues of Hecke–Maass cusp forms for the group $SL (n, ℤ)$ , with general $n \geq 3$ , over short intervals of certain length under the assumption of the generalised Lindelöf hypothesis and a slightly stronger upper bound concerning the exponent towards the Ramanujan–Petersson conjecture than is currently known. In particular, in this case we evaluate the second moment of the sums in question asymptotically.

Reçu le : 2019-12-30
Révisé le : 2020-06-19
Accepté le : 2020-08-10
Publié le : 2021-01-08

DOI : 10.5802/jtnb.1142

Classification : 11F12, 11F30
Mots clés : Fourier coefficients of automorphic forms, higher rank Maass cusp forms, sums of Hecke eigenvalues

Affiliations des auteurs :

Jesse Jääsaari ¹

¹ Department of Mathematics and Statistics FI-20014 University of Turku, Finland

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     journal = {Journal de th\'eorie des nombres de Bordeaux},
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     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Jesse Jääsaari. On Short Sums Involving Fourier Coefficients of Maass Forms. Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 3, pp. 761-793. doi : 10.5802/jtnb.1142. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1142/

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