MOYENNES DE FONCTIONS ARITHMÉTIQUES DE FORMES BINAIRES

Extending classical results of Nair and Tenenbaum, we provide general, sharp upper bounds for sums of the type∑u   < m   ⩽   u + vx   < n   ⩽   x + yF ( 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.