Given an arithmetic function $f$ such that the partial sums $\sum_{n \leq x} |f(n)|$ converge as $x$ approaches $\infty$, are there any results concerning the convergence properties of the series of sums of the form $\sum_{n \leq x} |f^{-1}(n)|$, where $f^{-1}$ is the Dirichlet inverse of $f$? Are there specific, known circumstances under which these sums will converge as well?