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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.

MSC:
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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