On Ranges of Variants of the Divisor Functions that are Dense

Abstract

For a real number $t$, let $s_t$ be the multiplicative arithmetic function defined by $s_t(p^{\hat{I}}\pm )=\sum _{j=0}^{\hat{I}}\pm (-p^t{)}^j$ for all primes $p$ and positive integers $\hat{I}\pm$. 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 $\hat{I}{\cdot }_A\approx 1.9011618$ and show that if $r>1$, then the range of the function $s_{-r}$ is a dense subset of the interval $\left(\frac{1}{\hat{I}\P (r)},1\right]$ if and only if $r\le \hat{I}{\cdot }_A$. We end with an open problem.

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