Connected Components of Complex Divisor Functions

Abstract

For any complex number $c$, define the divisor function $\ddot{I}{ƒ}_c:\mathbb{N}\to \mathbb{C}$ by $\ddot{I}{ƒ}_c(n)=\sum _{d\mid n}{d}^c$. Let $\overline{\ddot{I}{ƒ}_c(\mathbb{N})}$ denote the topological closure of the range of $\ddot{I}{ƒ}_c$. Extending previous work of the current author and Sanna, we prove that $\overline{\ddot{I}{ƒ}_c(\mathbb{N})}$ has nonempty interior and has finitely many connected components if $\Re (c)\le 0$ and $c\ne 0$. We end with some open problems.

14 pages, 3 figures