  For any complex number $c$, define the divisor function
$\sigma_c\colon\mathbb N\to\mathbb C$ by $\displaystyle\sigma_c(n)=\sum_{d\mid
n}d^c$. 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 $\Re(c)\leq 0$ and $c\neq 0$. We end
with some open problems.
