Sums involving the largest prime divisor of an integer. II (Q1113934)

This is the second part of the authors' paper [Acta Arith. 48, 1-8 (1987; Zbl 0578.10046)] on sums involving the largest prime divisor of an integer. A nice elementary method for the evaluation of sums of the form \(\sum_{2\leq n\leq x}f(n)P(n)\) or \(\sum_{2\leq n\leq x}f(n)g(n)\) is presented. Here P(n) is the largest prime divisor of n, f(n) and g(n) denote suitable multiplicative and additive functions, respectively. An analytic method for the evaluation of the latter sum has been developed by the reviewer [Publ. Inst. Math., Nouv. Sér. 41(55), 31-41 (1987; Zbl 0648.10028)], some of whose results are reproved now by the elementary method. The paper contains too many asymptotic formulas to be reproduced here. A typical one is \[ \sum_{2\leq n\leq x}\phi (n)P(n)=x^ 3\sum^{N}_{i=1}f_{i-1}(\log x)^{-i}+O(x^ 3(\log x)^{-N-1}), \] where \(\phi\) is the Euler function, the f's are suitable constants and \(N\geq 1\) is an arbitrary, but fixed integer. Reviewer's remark: In the paper \(O(x^ 3...)\) in the above formula is misprinted as \(O(x^ 2...)\). There are many other misprints: the binomial coefficient \(\left( \begin{matrix} k\\ i\end{matrix} \right)\) is printed as \(\left[ \begin{matrix} k\\ i\end{matrix} \right]\), in (3.19) and (3.21) \(\mu\) should be \(\phi\) and \(\sigma\) respectively, etc.