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We propose a generic algorithm for computing the inverses of a multiplicativefunction under the assumption that the set of inverses is finite. Moregenerally, our algorithm can compute certain functions of the inverses, such astheir power sums (e.g., cardinality) or extrema, without direct enumeration ofthe inverses. We illustrate our algorithm with Euler's totient function$\varphi(\cdot)$ and the $k$-th power sum of divisors $\sigma_k(\cdot)$. Forexample, we can establish that the number of solutions to $\sigma_1(x) =10^{1000}$ is 15,512,215,160,488,452,125,793,724,066,873,737,608,071,476, whileit is intractable to iterate over the actual solutions.

Computing the inverses, their power sums, and extrema for Euler's totient and other multiplicative functions