Sur la corrélation des fonctions de Piltz. (On the correlation of Piltz' divisor functions) (Q1080460)

Let \(d_ k(n)\) be the generalized divisor function. It is proved that for each \(k\geq 4\) there exists a polynomial \(p_ k\) of degree k, and a constant \(c_ k\) such that \[ \sum_{n\leq x}d_ k(n) d(n+1)=x p_ k(\log x)+O(x \exp (-c_ k(\log x)^{1/2})). \] The proof uses estimates of ''Bombieri-Vinogradov type'' taken from the first author's paper [J. Reine Angew. Math. 357, 51-76 (1985; Zbl 0547.10039)]. These estimates depend themselves on the Deshouillers-Iwaniec theory of averages of Kloosterman sums [cf. \textit{J.-M. Deshouillers} and \textit{H. Iwaniec}, Invent. Math. 70, 219-288 (1982; Zbl 0502.10021)]. It is somewhat disappointing that one does not obtain an error term \(O(x^{1- c})\) as one has when \(k\leq 3\).