  Let $\alpha$ and $\beta$ be two nonnegative integers such that $\beta <
\alpha$. 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 $\sum_{k\geqslant 1} S(\alpha k-\beta,n) a_k$, 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.
