Number theory
On small zeros of automorphic L-functions
[Petits zéros des fonctions L de formes automorphes]

Présenté par : Jean-Pierre Serre

Omar, Sami1
1 Faculty of Sciences of Tunis, Department of Mathematics, 2092 Campus universitaire El Manar Tunis, Tunisia
Comptes Rendus. Mathématique, Tome 352 (2014) no. 7-8, pp. 551-556.

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 GLd. 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.

In this paper, we first formulate the Weil explicit formula of prime number theory for cuspidal automorphic L-functions L(s,π) of GLd. 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.06.004
Omar, Sami 1

1 Faculty of Sciences of Tunis, Department of Mathematics, 2092 Campus universitaire El Manar Tunis, Tunisia 
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Omar, Sami. On small zeros of automorphic L-functions. Comptes Rendus. Mathématique, Tome 352 (2014) no. 7-8, pp. 551-556. doi : 10.1016/j.crma.2014.06.004. https://www.numdam.org/articles/10.1016/j.crma.2014.06.004/

[1] Barner, K. On A. Weil's explicit formula, J. Reine Angew. Math., Volume 323 (1981), pp. 139-152

[2] Blomer, V.; Brumley, F. The role of the Ramanujan conjecture in analytic number theory, Bull. Amer. Math. Soc., Volume 50 (2013), pp. 267-320

[3] Godement, R.; Jacquet, H. Zeta Functions of Simple Algebras, Lecture Notes in Mathematics, vol. 260, Springer-Verlag, Berlin, New York, 1972

[4] Iwaniec, H.; Kowalski, E. Analytic Number Theory, Colloquium Publications, vol. 53, American Mathematical Society, 2004

[5] Iwaniec, H.; Sarnak, P. Perspectives on the analytic theory of L-functions, Visions in Mathematics, 2000, pp. 705-741 Geom. Funct. Anal. (Special Volume – GAFA2000)

[6] Jacquet, H.; Shalika, J. On Euler products and the classification of automorphic representations, Amer. J. Math., Volume 103 (1981), pp. 499-588

[7] Luo, W.; Rudnik, Z.; Sarnak, P. On the generalized Ramanujan conjecture for GL(n), Proc. Sympos. Pure Math., vol. 66, part 2, 1999, pp. 301-310

[8] Mestre, J.-F. Formules explicites et minorations de conducteurs de variétés algébriques, Compos. Math., Volume 58 (1986), pp. 209-232

[9] Omar, S. Majoration du premier zéro de la fonction zêta de Dedekind, Acta Arith., Volume 95 (2000), pp. 61-65

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