On Ranges of Variants of the Divisor Functions that are Dense

TL;DR

This paper investigates the density of the ranges of certain divisor-related functions, showing they are dense in specific intervals depending on a parameter, and identifies a critical threshold for density when the parameter exceeds 1.

Contribution

It characterizes the density of the range of the function s_{-r} for different values of r, including a precise threshold for when the range is dense in a specific interval.

Findings

Range of s_{-r} is dense in (0,1] for r in (0,1].

For r > 1, the range is dense in (1/ζ(r), 1] if and only if r ≤ η_A ≈ 1.9011618.

Identifies a critical parameter η_A where the density property changes.

Abstract

For a real number $t$, let $s_t$ be the multiplicative arithmetic function defined by $\displaystyle{s_t(p^{\alpha})=\sum_{j=0}^{\alpha}(-p^t)^j}$ for all primes $p$ and positive integers $\alpha$. We show that the range of a function $s_{-r}$ is dense in the interval $(0,1]$ whenever $r\in(0,1]$. We then find a constant $\eta_A\approx1.9011618$ and show that if $r>1$, then the range of the function $s_{-r}$ is a dense subset of the interval $\displaystyle{\left(\frac{1}{\zeta(r)},1\right]}$ if and only if $r\leq \eta_A$. We end with an open problem.