On Short Sums Involving Fourier Coefficients of Maass Forms

Jääsaari, Jesse

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

Nous étudions les sommes des valeurs propres des opérateurs de Hecke des formes paraboliques de Hecke–Maass pour le groupe $\mathrm{SL}(n,\mathbb{Z})$ avec $n\ge 3$ 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 $\mathrm{SL}(n,\mathbb{Z})$, with general $n\ge 3$, 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.

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

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