# What is known about absolute convergence of Dirichlet inverses?

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?

• If $f$ is multiplicative, then $\sum|f(n)|<\infty$ is equivalent to $\sum|f^{-1}(n)|<\infty$. Indeed, an Euler product converges absolutely if and only if its reciprocal converges absolutely. Jan 14 at 20:53