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

We propose a generic algorithm for computing the inverses of a multiplicative\nfunction under the assumption that the set of inverses is finite. More\ngenerally, our algorithm can compute certain functions of the inverses, such as\ntheir power sums (e.g., cardinality) or extrema, without direct enumeration of\nthe inverses. We illustrate our algorithm with Euler's totient function\n$\\varphi(\\cdot)$ and the $k$-th power sum of divisors $\\sigma_k(\\cdot)$. For\nexample, we can establish that the number of solutions to $\\sigma_1(x) =\n10^{1000}$ is 15,512,215,160,488,452,125,793,724,066,873,737,608,071,476, while\nit is intractable to iterate over the actual solutions.\n