On the densities of covering numbers and 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 $0.103230<d(\mathcal{C})<\mathrm{0.103398.}$ Our approach adapts methods developed by Behrend and Deléglise 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 $c(n)\le h(n)=\sigma (n)/n$. 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 $O\left(x\exp \left(\left(-\frac{1}{2\sqrt{\log 2}}+\epsilon \right)\sqrt{\log x}\log \log x\right)\right)$, which is substantially smaller than the corresponding bound for primitive abundant numbers.