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

TL;DR

The paper introduces a generic algorithm to compute inverses and related functions for multiplicative functions like Euler's totient, enabling calculations of solutions counts without explicit enumeration.

Contribution

A novel algorithm that efficiently computes inverses, power sums, and extrema of multiplicative functions assuming finite inverse sets, demonstrated on Euler's totient and divisor functions.

Findings

Can determine the number of solutions to $\sigma_1(x) = 10^{1000}$ as 15,512,215,160,488,452,125,793,724,066,873,737,608,071,476

Enables computation of inverse-related functions without enumerating all solutions

Applicable to functions with finite inverse sets, broadening computational number theory tools

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.