Complex Divisor Functions

Abstract

For any complex number $c$, let $\ddot{I}{ƒ}_c:\mathbb{N}\to \mathbb{C}$ denote the divisor function defined by $\ddot{I}{ƒ}_c(n)=\sum _{d|n}{d}^c$ for all $n\in \mathbb{N}$, and define $R(c)=\{\ddot{I}{ƒ}_c(n)\in \mathbb{C}:n\in \mathbb{N}\}$ to be the range of $\ddot{I}{ƒ}_c$. We study the basic topological properties of the sets $R(c)$. In particular, we determine the complex numbers $c$ for which $R(c)$ is bounded and determine the isolated points of the sets $R(c)$. In the third section, we find those values of $c$ for which $R(c)$ is dense in $\mathbb{C}$. We also prove some results and pose several open problems about the closures of the sets $R(c)$ when these sets are bounded.

24 pages, 2 figures