Moyennes de fonctions arithmétiques de formes binaires

International audienceExtending classical results of Nair and Tenenbaum, we provide general, sharp upper bounds for sums of the type u<mu+v x<nx+y F (Q 1 (m, n),. .. , Q k (m, n)) where x, y, u, v have comparable logarithms, F belongs to a class defined by a weak form of sub-multiplicativity, and the Q j are arbitrary binary forms. A specific feature of the results is that the bounds are uniform within the F-class and that, as in a recent version given by Henriot, the dependency with respect to the coefficients of the Q j is made explicit. These estimates play a crucial rôle in the proof, published separately by the authors, of Manin's conjecture for Châtelet surfaces., count of lattice points over algebraic varieties. Résumé. Généralisant des résultats classiques de Nair et Tenenbaum, nous fournissons des majorations générales et optimales pour des sommes du type u<mu+v x