A class of sums involving the smallest prime factor of an integer (Q1337363)

Let \(h(0)=0\), \(h(n)= O(n^ k)\) for some \(k>0\), and define the additive function \(H(n)\) by \(H(1)=0\) and \(H(n)= \sum_{p^ a\parallel n} h(a)\) for \(n>1\). The author proves, using a nice elementary technique, that \[ \sum_{2\leq n\leq x} {{H(n)} \over {p(n)}}= C_ 0 h(1)x\log \log x+ C_ 1 x+ O(x/\log x), \] where \(p(n)\) is the smallest prime factor of \(n\), and \(C_ 0\), \(C_ 1\) are explicitly given constants with \(C_ 0>0\).