On Gaps in the Closures of Images of Divisor Functions

TL;DR

This paper investigates the topological structure of the closures of divisor function images for negative real exponents, establishing exponential lower bounds on their connected components and analyzing their gap structures.

Contribution

It introduces exponential lower bounds on the number of connected components of the closures of divisor function images for r>1, advancing previous linear bounds.

Findings

Exponential lower bounds on connected components

Analysis of gap structures in closures

Insights towards monotonicity of the closures

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.