On the Number of Connected Components of Ranges of Divisor Functions

TL;DR

This paper investigates the topological structure of the range of divisor functions, establishing bounds on the number of connected components and demonstrating that this number cannot be certain integers, such as four.

Contribution

It provides explicit bounds on the number of connected components of the divisor function range and shows certain values are impossible for this count.

Findings

Bounds on the number of connected components of the divisor function range.

The number of components is at least rom primes up to r.

The number of components cannot be exactly 4.

Abstract

For $r \in \mathbb{R}, r> 1$ and $n \in \mathbb{Z}^+$, the divisor function $\sigma_{-r}$ is defined by $\sigma_{-r}(n) := \sum_{d \vert n} d^{-r}$. In this paper we show the number $C_r$ of connected components of $\overline{\sigma_{-r}(\mathbb{Z}^+)}$ satisfies $$\pi(r) + 1 \leq C_r \leq \frac{1}{2}\exp\left[\frac{1}{2}\dfrac{r^{20/9}}{(\log r)^{29/9}} \left( 1 + \frac{\log\log r}{\log r - \log\log r} + \frac{\mathcal{O}(1)}{\log r}\right) \right],$$ where $\pi(t)$ is the number of primes $p\leq t$. We also show that $C_r$ does not take all integer values, specifically that it cannot be equal to $4$.