Let \(\nu_y(n)\) denote the number of distinct prime factors of \(n\) that are \(<y\) . For k a positive integer, and for \(k+ 2 \leq y \leq x\) ,let \(S_{−k}(x,y)\) denote the sum \(S_{−k}(x,y)=\sum_{n\leq x}(-k)^{\nu_y(n)}\) . We will describe our recent results on the asymptotic behavior of \(S_{−k}(x,y)\) for \(k+ 2 \leq y \leq x\) , and \(x\) sufficiently large. There is a crucial difference in the asymptotic behavior of \(S_{−k}(x,y)\) when \(k+ 1\) is a prime and \(k+ 1\) is composite, and this makes the problem particularly interesting. The results are derived utilizing a combination of the Buchstab-de Bruijn recurrence, the Perron contour integral method, and certain difference-differential equations. We present our results against the background of earlier work of the first author on sums of the Möbius function over integers with restricted prime factors and on a mutiplicative generalization of the sieve. This is joint work with my former PhD student Ankush Goswami.