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Complex Divisor Functions
Defant, Colin...

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Complex Divisor Functions by Defant, Colin ( Author )
Lemonjar Open Access
Australian National University
07-09-2023
For any complex number $c$, let $\sigma_c\colon\mathbb N\rightarrow\mathbb C$ denote the divisor function defined by $\sigma_c(n)=\displaystyle{\sum_{d|n}d^c}$ for all $n\in\mathbb N$, and define $R(c)=\{\sigma_c(n)\in\mathbb C\colon n\in\mathbb N\}$ to be the range of $\sigma_c$. 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.
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Article
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29.34 KB
English
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MYR 0.01
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http://arxiv.org/abs/1502.03417
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