On Gaps in the Closures of Images of Divisor Functions

Abstract

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.

10 pages