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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6$\begingroup$ If $f$ is multiplicative, then $\sumf(n)<\infty$ is equivalent to $\sumf^{1}(n)<\infty$. Indeed, an Euler product converges absolutely if and only if its reciprocal converges absolutely. $\endgroup$– GH from MOJan 14, 2021 at 20:53
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