On Ranges of Variants of the Divisor Functions that are Dense

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 ${\eta }_A\approx 1.9011618$ and show that if $r>1$, then the range of the function $s_{-r}$ is a dense subset of the interval ${\displaystyle (\frac{1}{\zeta (r)},1]}$ if and only if $r\leq \eta_A$. We end with an open problem.