On Gaps in the Closures of Images of Divisor Functions

Given a complex number $c$, define the divisor function $\sigma_c:\mathbb N\to\mathbb C$ by $\sigma_c(n)=\sum_{d\mid n}d^c$. In this paper, we look at $\overline{\sigma_{-r}(\mathbb N)}$, the topological closures of the image of $\sigma_{-r}$, when $r>1$. We exhibit new lower bounds on the number of connected components of $\overline{\sigma_{-r}(\mathbb N)}$, bringing this bound from linear in $r$ to exponential. Finally, we discuss the general structure of gaps of $\overline{\sigma_{-r}(\mathbb N)}$ in order to work towards a possible monotonicity result.