On the densities of covering numbers and abundant numbers

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.

💡 Research Summary

The paper investigates two classical sets of integers: covering numbers (C) and abundant numbers (A). A covering number n is defined as an integer for which there exists a distinct covering system— a collection of congruences (a_i mod m_i) with pairwise different moduli— such that every modulus m_i divides n and the system covers all integers. The smallest covering number is 12, and the property is multiplicative: any multiple of a covering number is again a covering number. Primitive covering numbers are those not divisible by a smaller covering number. An abundant number satisfies σ(n) > 2n, where σ is the sum‑of‑divisors function; primitive abundant numbers are abundant but have no proper abundant divisor.

The authors first prove that the set C possesses a natural density d(C). To establish existence they adapt Erdős’s classic argument: the density exists iff the sum of reciprocals of primitive elements converges. Using a new inequality (Lemma 3.1) they show that for any primitive covering number n the largest prime factor P⁺(n) is bounded by the number of divisors of n/P⁺(n). This forces primitive covering numbers to be extremely sparse. By combining known estimates for smooth numbers ψ(x,y) (De Bruijn) and for numbers with many divisors Δ(x,y) (Norton), and choosing y = exp(√log 2·√log x), they obtain an upper bound \