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 $\displaystyle{s_t(p^α)=\sum_{j=0}^α(-p^t)^j}$ for all primes $p$ and positive integers $α$. 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 $η_A\approx1.9011618$ and show that if $r>1$, then the range of the function $s_{-r}$ is a dense subset of the interval $\displaystyle{\left(\frac{1}{ζ(r)},1\right]}$ if and only if $r\leq η_A$. We end with an open problem.

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