On the densities of covering numbers and abundant numbers

TL;DR

This work establishes that the set of covering numbers $al C$ has a natural density with explicit numerical bounds $0.103230<d(al C)<0.103398$, and it simultaneously sharpens the known density bounds for abundant numbers $al A$ to $0.247619608<d(al A)<0.247619658$. The authors adapt the Behrend–Deléglise approach, introducing the function $c(n)$ to measure proximity to covering behavior and developing a computable upper bound $c'(n)$ via Sun-almost-covering divisors to make large-scale calculations tractable. They provide a refined framework using almost-covering numbers, complementary Bell numbers, and a generalized Stirling bound to bound densities and to bound the counting function of primitive covering numbers by $O\left(x\exp\left(\left(-\tfrac{1}{2\sqrt{\log 2}}+\varepsilon\right)\sqrt{\log x}\log\log x\right)\right)$. A substantial computational component then delivers explicit numerical bounds and a practical lower bound for $d(al C)$ by enumerating small primitive covering numbers, illustrating both the depth of the theory and its computability. The results deepen the connection between covering and abundant numbers and provide tools for precise density estimation in related arithmetic-structure problems.

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 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 $c(n) \leq h(n) = σ(n)/n$. 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 $O\left( x\exp\left(\left(-\tfrac{1}{2\sqrt{\log 2}} + ε\right)\sqrt{\log x} \log \log x\right)\right)$, which is substantially smaller than the corresponding bound for primitive abundant numbers.

Theorems & Definitions (35)

• Theorem 1.1: Sun
• Conjecture 1.2: Erdős-Selfridge
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Corollary 2.4
• Theorem 2.5
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['thm:primCovNums']}