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 ${\sigma }_{-r}(n):={\sum }_{d\mathrm{\mid }n}{d}^{-r}$. In this paper we show the number $C_r$ of connected components of $\overline{{\sigma }_{-r}({\mathbb{Z}}^{+})}$ satisfies $$\pi (r)+1\le C_r\le \frac{1}{2}\exp \left[\frac{1}{2}{\displaystyle \frac{r^{20\mathrm{/}9}}{(\log r{)}^{29\mathrm{/}9}}}(1+\frac{\log \log r}{\log r-\log \log r}+\frac{\mathcal{O}(1)}{\log r})\right],$$ where $\pi (t)$ is the number of primes $p\le t$. We also show that $C_r$ does not take all integer values, specifically that it cannot be equal to $4$.