[Paper Review] Complex Divisor Functions

Colin Defant|arXiv (Cornell University)|Feb 11, 2015

Mathematical Dynamics and Fractals2 references3 citations

TL;DR

This paper investigates the topological properties of the range $ R(c) = \{\sigma_c(n) \mid n \in \mathbb{N}\} $, where $ \sigma_c(n) = \sum_{d|n} d^c $ for complex $ c $. It determines when $ R(c) $ is bounded, identifies its isolated points, and characterizes when $ R(c) $ is dense in $ \mathbb{C} $, showing that for $ \operatorname{Re}(c) < -3.02 $, the closure of $ R(c) $ is disconnected, and for certain $ c $, $ R(c) $ is dense in the complex plane.

ABSTRACT

For any complex number $c$, let $σ_c\colon\mathbb N ightarrow\mathbb C$ denote the divisor function defined by $σ_c(n)=\displaystyle{\sum_{d|n}d^c}$ for all $n\in\mathbb N$, and define $R(c)=\{σ_c(n)\in\mathbb C\colon n\in\mathbb N\}$ to be the range of $σ_c$. 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.

Motivation & Objective

• To analyze the topological structure of the range $ R(c) = \{\sigma_c(n) \mid n \in \mathbb{N}\} $ for complex $ c $, extending beyond classical divisor functions.
• To determine for which complex $ c $ the set $ R(c) $ is bounded, has isolated points, or is dense in $ \mathbb{C} $.
• To investigate the closure $ \overline{R(c)} $, especially its connectedness and fractal-like behavior, and to identify open problems in this area.

Proposed method

• Uses complex analysis and properties of multiplicative arithmetic functions to study $ \sigma_c(n) = \sum_{d|n} d^c $, particularly for $ c = a + ib $.
• Applies symmetry via $ \sigma_{\overline{c}}(n) = \overline{\sigma_c(n)} $ to reduce analysis to the upper half-plane.
• Employs asymptotic estimates and inequalities involving $ \zeta(s) $, $ \sigma_c(p^\alpha) = \frac{p^{(\alpha+1)c} - 1}{p^c - 1} $, and trigonometric identities for complex arguments.
• Uses numerical and analytical bounds on $ F(x) $ to show that $ \overline{R(c)} $ is disconnected when $ \operatorname{Re}(c) < -3.02 $, based on separation of $ \sigma_c(S_1) $ and $ \sigma_c(V_1) $.
• Leverages results from prior work on real $ c $, especially the constant $ \eta \approx 1.8877909 $, to extend density results to complex $ c $.
• Uses Mathematica visualizations to observe fractal-like patterns in $ R(c) $, informing conjectures about closure structure.

Experimental results

Research questions

• RQ1 For which complex numbers $ c $ is the range $ R(c) $ bounded?
• RQ2 Which values of $ c $ result in $ R(c) $ having isolated points?
• RQ3 For which $ c \in \mathbb{C} $ is $ R(c) $ dense in the entire complex plane $ \mathbb{C} $?
• RQ4 When is the closure $ \overline{R(c)} $ connected, and what conditions lead to its disconnection?
• RQ5 What is the geometric structure of $ \overline{R(c)} $, and how does it relate to $ \zeta(1 - c) $?

Key findings

• The set $ R(c) $ is bounded if and only if $ \operatorname{Re}(c) < -1 $, with the closure $ \overline{R(c)} $ being disconnected when $ \operatorname{Re}(c) < -3.02 $.
• The range $ R(c) $ contains 0 if and only if $ \operatorname{Re}(c) = 0 $ and $ \operatorname{Im}(c) = q \frac{\pi}{\log p} $ for some prime $ p $ and rational $ q $ not an even integer.
• For $ c $ with $ \operatorname{Re}(c) = 0 $, the set $ \{\sigma_c(p) \mid p \in \mathbb{P}\} $ is dense in the unit circle $ \{1 + z \mid |z| = 1\} $, implying $ R(c) $ is dense in $ \mathbb{C} $ under certain conditions.
• When $ c $ is real and $ -\kappa < c < -\eta $ with $ \eta \approx 1.8877909 $, $ \overline{R(c)} $ is the disjoint union of two connected sets, indicating a phase transition in structure.
• For $ c = -1.3 + i $, the point $ \zeta(1 - c) $ is visually distinct from the centroid of $ \overline{R(c)} $, suggesting non-uniform density in the range.
• The paper identifies open problems regarding the exact characterization of $ \mathcal{A}(c) $ and $ C(c) $, and calls for improved bounds on $ \Theta(c) $, indicating ongoing research potential.