On sums formed with the largest prime divisor of an integer (Q1337756)

Let \(\mathbb{Q}\) be a non-empty set of primes such that \[ \sum_{p\leq x, p\in \mathbb{Q}} \log p=\delta x+ O(x(\log x)^{-A}) \] for some constants \(0\leq\delta \leq 1\) and \(A>0\). If \(N(\mathbb{Q})\) is the set of all natural numbers which have a prime divisor from \(\mathbb{Q}\), then the author establishes an asymptotic formula for \(\sum_{n\leq x, n\in N(\mathbb{Q})} f(P(n, \mathbb{Q}))\), where \(P(n, \mathbb{Q})\) is the largest prime factor of \(n\) belonging to \(\mathbb{Q}\), and \(f\) satisfies certain conditions. The paper is motivated by the works of \textit{J.-M. De Koninck} [Monatsh. Math. 116, 13- 37 (1993; Zbl 0788.11039)] and \textit{J.-M. De Koninck} and the reviewer [On some asymptotic formulas related to large additive functions over primes of positive density, Math. Balkanica 8 (1994) (in press)] where the cases \(f(y)= 1/y\) and \(f(y)= 1/\log y\) were treated. The author obtains better error terms than the ones obtained in those papers.