Connected Components of Complex Divisor Functions

TL;DR

This paper investigates the topological structure of the divisor function's range for complex exponents, proving it has nonempty interior and finitely many connected components under certain conditions.

Contribution

It extends previous work by characterizing the connected components of the divisor function's range for complex parameters with nonpositive real part.

Findings

The closure of the divisor function's range has nonempty interior.

It has finitely many connected components when the real part of c is less than or equal to zero, excluding zero.

The paper discusses open problems related to the divisor function.

Abstract

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.