On Gaps in the Closures of Images of Divisor Functions

Niven Achenjang; Aaron Berger

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arXiv · 1808.07550

On Gaps in the Closures of Images of Divisor Functions

Niven Achenjang·Aaron Berger

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Abstract

Given a complex number $c$, define the divisor function $\sigma_c:\mathbb N\to\mathbb C$ by $\sigma_c(n)=\sum_{d\mid n}d^c$. In this paper, we look at $\overline{\sigma_{-r}(\mathbb N)}$, the topological closures of the image of $\sigma_{-r}$, when $r>1$. We exhibit new lower bounds on the number of connected components of $\overline{\sigma_{-r}(\mathbb N)}$, bringing this bound from linear in $r$ to exponential. Finally, we discuss the general structure of gaps of $\overline{\sigma_{-r}(\mathbb N)}$ in order to work towards a possible monotonicity result.

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Niven Achenjang, Aaron Berger. 2018-08-22. On Gaps in the Closures of Images of Divisor Functions. https://arxiv.org/abs/1808.07550

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