Multiplicative Functions and Small Divisors

Let S be a set of positive integers and g be a multiplicative function. Consider the problem of estimating the sum 1.1$${\text{S}}\left( {{\text{x,g}}} \right) = \sum\limits_{\mathop {n \leqslant x}\limits_{n \in S} } {g\left( {\text{n}} \right).} {\text{ }}$$ A natural way to start is to write 1.2$$\text{g}\left( \text{n} \right)\text{ = }\sum\limits_{\text{d}\left| \text{n} \right.} {\text{h}\left( \text{d} \right)}$$ and reverse the order of summation. This in turn leads to the estimation of the contribution arising from the large divisors d of n, where n S, which often presents difficulties.