On the densities of covering numbers and abundant numbers

TL;DR

This paper determines the natural density of covering numbers with high precision, introduces a new method to estimate densities of abundant and covering numbers, and improves bounds for the density of abundant numbers.

Contribution

It establishes the density of covering numbers, refines bounds for abundant numbers, and develops new computational techniques for these density estimates.

Findings

Density of covering numbers is approximately 0.10323 to 0.10340.

Bounds for the density of abundant numbers are tightened.

Count of primitive covering numbers up to x is significantly smaller than for abundant numbers.

Abstract

We investigate the densities of the sets of abundant numbers and of covering numbers, integers $n$ for which there exists a distinct covering system where every modulus divides $n$. We establish that the set $\mathcal{C}$ of covering numbers possesses a natural density $d(\mathcal{C})$ and prove that $0.103230 < d(\mathcal{C}) < 0.103398.$ Our approach adapts methods developed by Behrend and Del\'eglise for bounding the density of abundant numbers, by introducing a function $c(n)$ that measures how close an integer $n$ is to being a covering number with the property that $c(n) \leq h(n) = \sigma(n)/n$. However, computing $d(\mathcal{C})$ to three decimal digits requires some new ideas to simplify the computations. As a byproduct of our methods, we obtain significantly improved bounds for $d(\mathcal{A})$, the density of abundant numbers, namely $0.247619608 < d(\mathcal{A}) < 0.247619658$. We also show the count of primitive covering numbers up to $x$ is $O\left( x\exp\left(\left(-\tfrac{1}{2\sqrt{\log 2}} + \epsilon\right)\sqrt{\log x} \log \log x\right)\right)$, which is substantially smaller than the corresponding bound for primitive abundant numbers.