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Comptes Rendus Mathématique
Volume 353, n° 8
pages 677-682 (août 2015)
Doi : 10.1016/j.crma.2015.04.018
Received : 19 January 2015 ;  accepted : 22 April 2015
A simple proof of the mean value of   in function fields
Une démonstration simple de la valeur moyenne de   dans des corps de fonctions
 

Julio Andrade a, b
a Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK 
b Depto. Matematica, PUC-Rio, Rio De Janeiro, RJ, Brazil 

Abstract

Let F be a finite field of odd cardinality q ,   the polynomial ring over F ,   the rational function field over F and   the set of square-free monic polynomials in A of degree odd. If  , we denote by   the integral closure of A in  . In this Note, we give a simple proof for the average value of the size of the groups   as D varies over the ensemble   and q is kept fixed. The proof is based on character sums estimates and on the use of the Riemann hypothesis for curves over finite fields.

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Résumé

Soit F un corps fini de cardinalité impaire q ,   l'anneau de polynômes sur F ,   le corps des fonctions rationnelles sur F et   l'ensemble des polynômes unitaires et sans facteur carré en A de degré impair. Si  , on note par   la clóture intégrale de A en  . Dans cette Note, nous donnons une preuve simple de la valeur moyenne de la taille des groupes   quand D varie dans l'ensemble   et quand q est maintenu fixe. La preuve est basée sur des estimations des sommes de caractères et sur l'utilisation de l'hypothèse de Riemann pour les courbes sur les corps finis.

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