Divisor Functions: Train-like Structure and Density Properties

TL;DR

This paper explores the density and structural properties of generalized divisor functions, extending known results for s=1 to all s≥0, revealing train-like structures and density thresholds, with implications for number theory and distribution of rational numbers.

Contribution

It introduces the concept of trains in divisor functions, extends Wolke's conjecture results, and characterizes the density and range properties of these functions across different s values.

Findings

f_s is locally dense for s>0

f_s is dense for 0<s≤1 but not for s>1

the rational complement of the range is dense for all s>0

Abstract

We investigate the density properties of generalized divisor functions $\displaystyle f_s(n)=\frac{\sum_{d|n}d^s}{n^s}$ and extend the analysis from the already-proven density of $s=1$ to $s\geq0$. We demonstrate that for every $s>0$, $f_s$ is locally dense, revealing the structure of $f_s$ as the union of infinitely many $trains$ -- specially organized collections of decreasing sequences -- which we define. We analyze Wolke's conjecture that $|f_1(n)-a|<\frac{1}{n^{1-\varepsilon}}$ has infinitely many solutions and prove it for points in the range of $f_s$. We establish that $f_s$ is dense for $0<s\leq1$ but loses density for $s>1$. As a result, in the latter case the graphs experience ruptures. We extend Wolke's discovery $\displaystyle |f_1(n)-a|<\frac{1}{n^{0.4-\varepsilon}}$ to all $0<s\leq1$. In the last section, we prove that the rational complement to the range of $f_s$ is dense for all $s>0$. Thus, the range of $f_1$ and its complement form a partition of rational numbers to two dense subsets. If we treat the divisor function as a uniformly distributed random variable, then its expectation turns out to be $\zeta(s+1)$. The theoretical findings are supported by computations. Ironically, perfect and multiperfect numbers do not exhibit any distinctive characteristics for divisor functions.