Dirichlet twists of $\protect \mathrm{GL}_n$-automorphic $L$-functions and hyper-Kloosterman Dirichlet series

Jeanine Van Order

doi:10.5802/afst.1687

Dirichlet twists of

{GL}_{n}

-automorphic

L

-functions and hyper-Kloosterman Dirichlet series

Jeanine Van Order ¹

¹ Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 3, pp. 633-703.

Abstract
Abstract

We calculate mean values of ${GL}_{n}$ -automorphic $L$ -functions twisted by primitive even Dirichlet characters of prime-power conductor, at arbitrary points within the critical strip, by derivation of special Voronoi summation formulae. Our calculation is novel in that the twisted sum can be expressed in terms of the average itself, and also that it sees the derivation of various new summation formulae in the setting of prime-power modulus. One consequence, as we explain, is to show the analytic continuation and additive summation formulae for hyper-Kloosterman Dirichlet series associated to ${GL}_{n}$ -automorphic $L$ -functions.

Nous calculons les valuers moyennes des fonctions $L$ automorphes sur ${GL}_{n}$ tordues par des caractères de Dirichlet primitifs et pairs, du conducteur une puissance d’un nombre premier, à des points arbitraires dans la bande critique, en dérivant des formules de sommation spéciales du type Voronoi. Notre calcul est nouveau car la somme est exprimé en termes de la moyenne elle-même, et aussi qu’il voit la dérivation de diverses nouvelles formules de sommation dans le regime des puissances d’un nombre premier. Une conséquence, comme nous l’expliquons, est de montrer les prolongations analytiques et des formules de sommation additive pour les séries de Dirichlet hyper-Kloosterman associées aux fonctions $L$ automorphes sur ${GL}_{n}$ .

Received: 2019-04-20
Accepted: 2019-12-13
Published online: 2021-10-26

DOI: 10.5802/afst.1687

Author's affiliations:

Jeanine Van Order ¹

¹ Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Jeanine Van Order},
     title = {Dirichlet twists of $\protect \mathrm{GL}_n$-automorphic $L$-functions and {hyper-Kloosterman} {Dirichlet} series},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {633--703},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 30},
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     doi = {10.5802/afst.1687},
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Jeanine Van Order. Dirichlet twists of $\protect \mathrm{GL}_n$-automorphic $L$-functions and hyper-Kloosterman Dirichlet series. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 3, pp. 633-703. doi : 10.5802/afst.1687. https://afst.centre-mersenne.org/articles/10.5802/afst.1687/

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