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For a real number $t$, let $s_t$ be the multiplicative arithmetic functiondefined by $\displaystyle{s_t(p^{\alpha})=\sum_{j=0}^{\alpha}(-p^t)^j}$ for allprimes $p$ 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 aconstant $\eta_A\approx1.9011618$ and show that if $r>1$, then the range of thefunction $s_{-r}$ 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.

On Ranges of Variants of the Divisor Functions that are Dense