Ranges of Unitary Divisor Functions

TL;DR

This paper investigates the topological properties of the ranges of unitary divisor functions, providing explicit constants and characterizations of their connectedness based on a parameter r.

Contribution

It explicitly calculates a critical constant and characterizes when the closure of the range of the unitary divisor function is connected.

Findings

Calculated an explicit constant η* ≈ 1.9742550.

Connectedness of the range closure depends on r in (0, η*].

Open problem posed regarding the structure of the range.

Abstract

For any real $t$, the unitary divisor function $\sigma_t^*$ is the multiplicative arithmetic function defined by $\sigma_t^*(p^{\alpha})=1+p^{\alpha t}$ for all primes $p$ and positive integers $\alpha$. Let $\overline{\sigma_t^*(\mathbb N)}$ denote the topological closure of the range $\sigma_t^*$. We calculate an explicit constant $\eta^*\approx 1.9742550$ and show that $\overline{\sigma_{-r}^*(\mathbb N)}$ is connected if and only if $r\in(0,\eta^*]$. We end with an open problem.