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April 10, 2015 By

A non-vanishing theorem on Dirichlet series

C. R. Math. Rep. Acad. Sci. Canada Vol. 27, (3), 2005 pp. 76–83

September 30, 2005

Wentang Kuo, Department of Pure Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1; email: wtkuo@math.uwaterloo.ca

Abstract/Résumé:

The non-vanishing property of certain Dirichlet series is a fundamental problem in analytic number theory. In this paper, we provide a non-vanishing theorem, which is a generalization of Ogg’s result. We apply our theorem to get applications on distributions of eigenvalues of Hecke eigenforms and recover the non-vanishing theorem for the \(L\)-functions of cuspidal representations.

La propriété non nulle de certaines séries de Dirichlet est un problème fondamental dans la théorie analytique des nombres. Dans cet article, nous fournissons un théorème non-non-vanishing, qui est une généralisation du résultat d’Ogg. Nous appliquons notre théorème pour obtenir des applications sur des distributions des valeurs propres des opérateurs de Hecke et nous récupèrous théorème non nulle pour les \(L\)-fonctions des représentations cuspidales.

Keywords: L-functions, elliptic curves, non-vanishing.
AMS Subject Classification: Modular and automorphic functions 11F03

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