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Mean values of arithmetic functions of binary forms. (Moyennes de fonctions arithmétiques de formes binaires.) (French. English summary) Zbl 1284.11126
Summary: Extending classical results of M. Nair and G. Tenenbaum [Acta Math. 180, No. 1, 119–144 (1998; Zbl 0917.11048)], we provide general, sharp upper bounds for sums of the type \[ \sum _{\substack{ {u<m\leq u+v} \\ {x<n\leq x+y} }} F(Q_1(m,n),\dots , Q_k(m,n)) \] 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 K. Henriot [Math. Proc. Camb. Philos. Soc. 152, No. 3, 405–424 (2012; Zbl 1255.11048)], 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.

11N37 Asymptotic results on arithmetic functions
11N56 Rate of growth of arithmetic functions
11A25 Arithmetic functions; related numbers; inversion formulas
11D45 Counting solutions of Diophantine equations
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