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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 


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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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.

The full text of this article is available in PDF format.

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