Let $α$ and $β$ be two nonnegative integers such that $β< α$. For an arbitrary sequence $\{a_n\}_{n\geqslant 1}$ of complex numbers, we consider the generalized Lambert series in order to investigate linear combinations of the form $\sum_{k\geqslant 1} S(αk-β,n) a_k$, where $S(k,n)$ is the total number of non-overlined parts equal to $k$ in all the overpartitions of $n$. The general nature of the numbers $a_n$ allows us to provide connections between overpartitions and functions from multiplicative number theory.