A new method for lower bounds of <i>L</i>-functions

Gelbart, Stephen S.; Lapid, Erez M.; Sarnak, Peter

doi:10.1016/j.crma.2004.04.024

Number Theory

A new method for lower bounds of L-functions
[Une nouvelle méthode pour minorer des fonctions L.]

Présenté par : Jean-Pierre Serre

Gelbart, Stephen S.¹ ; Lapid, Erez M.² ; Sarnak, Peter ^3,⁴

¹ Faculty of Mathematics and Computer Science, Nicki and J. Ira Harris Professorial Chair, The Weizmann Institute of Science, Rehovot 76100, Israel
² Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
³ Department of Mathematics, Princeton University, Princeton, NJ, USA
⁴ The Courant Institute, New York, NY, USA

Comptes Rendus. Mathématique, Tome 339 (2004) no. 2, pp. 91-94.

Résumé
Abstract

Soit $L (s, π, r)$ une fonction L présente dans la théorie de Langlands–Shahidi. Nous prouvons une minoration de $L (s, π, r)$ quand $R (s) = 1$ , en utilisant les séries d'Eisenstein. Cette méthode s'applique même lorsqu'on ne sait pas que $L (s, π, r)$ est absolument convergente pour $R (s) > 1$ .

Let $L (s, π, r)$ be an L-function which appears in the Langlands–Shahidi theory. We give a lower bound for $L (s, π, r)$ when $R (s) = 1$ using Eisenstein series. This method is applicable even when $L (s, π, r)$ is not known to be absolutely convergent for $R (s) > 1$ .

Reçu le : 2004-04-17
Accepté le : 2004-04-27
Publié le : 2004-07-15

DOI : 10.1016/j.crma.2004.04.024

Affiliations des auteurs :

Gelbart, Stephen S. ¹ ; Lapid, Erez M. ² ; Sarnak, Peter ^{3, 4}

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     author = {Gelbart, Stephen S. and Lapid, Erez M. and Sarnak, Peter},
     title = {A new method for lower bounds of {\protect\emph{L}-functions}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {91--94},
     publisher = {Elsevier},
     volume = {339},
     number = {2},
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Gelbart, Stephen S.; Lapid, Erez M.; Sarnak, Peter. A new method for lower bounds of L-functions. Comptes Rendus. Mathématique, Tome 339 (2004) no. 2, pp. 91-94. doi : 10.1016/j.crma.2004.04.024. http://www.numdam.org/articles/10.1016/j.crma.2004.04.024/

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