On small zeros of automorphic <i>L</i>-functions

Sami Omar

doi:10.1016/j.crma.2014.06.004

Number theory

On small zeros of automorphic L-functions

Presented by: Jean-Pierre Serre

Sami Omar ¹

¹ Faculty of Sciences of Tunis, Department of Mathematics, 2092 Campus universitaire El Manar Tunis, Tunisia

Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 551-556.

Abstract
Résumé

In this paper, we first formulate the Weil explicit formula of prime number theory for cuspidal automorphic L-functions $L (s, π)$ of ${GL}_{d}$ . Then, we prove some conditional results about the vanishing order at the central point of $L (s, π)$ . This enables to yield an estimate for the height of the lowest zero of $L (s, π)$ on the critical line in terms of the analytic conductor.

Dans cet article, nous formulons d'abord les formules explicites de Weil de la théorie des nombres premiers pour les fonctions L de formes automorphes cuspidales $L (s, π)$ de ${GL}_{d}$ . Ensuite, nous montrons des résultats conditionnels concernant l'ordre d'annulation de $L (s, π)$ au point $s = 1 / 2$ , ce qui permet de donner une estimation de la hauteur du plus petit zéro de $L (s, π)$ sur la droite critique en termes de conducteur analytique.

Received: 2014-03-10
Accepted: 2014-06-06
Published online: 2014-06-30

DOI: 10.1016/j.crma.2014.06.004

Author's affiliations:

Sami Omar ¹

¹ Faculty of Sciences of Tunis, Department of Mathematics, 2092 Campus universitaire El Manar Tunis, Tunisia

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     title = {On small zeros of automorphic {\protect\emph{L}-functions}},
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Sami Omar. On small zeros of automorphic L-functions. Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 551-556. doi : 10.1016/j.crma.2014.06.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.06.004/

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