On some sums involving the largest prime divisor of \(n\). II (Q2717633)

This is a continuation of the author's earlier work on the same subject [Acta Arith. 59, 339-363 (1991; Zbl 0708.11047) and Proc. Conf. in Honor of H. Halberstam, Birkhäuser, Boston, Prog. Math. 139, 723-735 (1996; Zbl 0853.11081)]. The focus is on asymptotic formulas of the form NEWLINE\[NEWLINE \Sigma_f = \sum_{nP(n)\leq x, n>1}1,\quad \Sigma_f'(x) = \sum_{1<n\leq x} {1\over f(P(n))}, NEWLINE\]NEWLINE where \(P(n)\) is the largest prime factor of \(n > 1\), and \(f\) is a positive increasing function satisfying some rather general conditions, such as NEWLINE\[NEWLINE \log f(w) = (\log w)^{\nu+o(1)}\qquad(\nu \geq 0, w\to\infty). NEWLINE\]NEWLINE In Theorem 1 the asymptotic formula for \(\Sigma_f(x)\) is derived, where the error term has a different shape when \(\nu \geq 3/5\) and \(3/8 \leq \nu < 3/5\); when \(0 \leq \nu < 3/8\) there is even a second main term. In the case when one assumes the Riemann hypothesis, one can obtain sharper results. These are furnished by Theorem 2, while Theorem 3 says that the first two theorems remain valid if \(\Sigma_f(x)\) is replaced by \(\Sigma'_f(x)\) and the main terms by appropriate new main terms. Since NEWLINE\[NEWLINE \Sigma_f(x) = \sum_{pf(p)\leq x}\psi\left({x\over pf(p)},p\right), \qquad \psi(x,y) = \sum_{n\leq x, P(n)\leq y}1, NEWLINE\]NEWLINE it is natural that the proofs of the results depend on intricate properties of the \(\psi(x,y)\)-function.