On the Number of Connected Components of Ranges of Divisor Functions

For $r \in \mathbb{R}, r> 1$ and $n \in \mathbb{Z}^+$, the divisor function $\sigma_{-r}$ is defined by $\sigma_{-r}(n) := \sum_{d \vert n} d^{-r}$. In this paper we show the number $C_r$ of connected components of $\overline{\sigma_{-r}(\mathbb{Z}^+)}$ satisfies $$\pi(r) + 1 \leq C_r \leq \frac{1}{2}\exp\left[\frac{1}{2}\dfrac{r^{20/9}}{(\log r)^{29/9}} \left( 1 + \frac{\log\log r}{\log r - \log\log r} + \frac{\mathcal{O}(1)}{\log r}\right) \right],$$ where $\pi(t)$ is the number of primes $p\leq t$. We also show that $C_r$ does not take all integer values, specifically that it cannot be equal to $4$.