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?

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    $\begingroup$ 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. $\endgroup$
    – GH from MO
    Jan 14, 2021 at 20:53


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