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

Abstract

We propose a generic algorithm for computing the inverses of a multiplicative function under the assumption that the set of inverses is finite. More generally, our algorithm can compute certain functions of the inverses, such as their power sums (e.g., cardinality) or extrema, without direct enumeration of the inverses. We illustrate our algorithm with Euler's totient function $\varphi(\cdot)$ and the $k$-th power sum of divisors $\sigma_k(\cdot)$. For example, 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, while it is intractable to iterate over the actual solutions.