Connected Components of Complex Divisor Functions

For any complex number $c$, define the divisor function ${\sigma }_c\text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{C}$ by $\displaystyle\sigma_c(n)=\sum_{d\mid n}d^c$\mathrm{.}Let$\overline{\sigma_c(\mathbb N)}$ denote the topological closure of the range of ${\sigma }_c$. Extending previous work of the current author and Sanna, we prove that $\overline{{\sigma }_c(\mathbb{N})}$ has nonempty interior and has finitely many connected components if $\mathrm{\Re }(c)\le 0$ and $c\mathrm{\ne }0$. We end with some open problems.