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 C of covering numbers possesses a natural density d(C) and prove that 0.103230 < d(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) ≤ h(n) = σ(n)/n. However, computing d(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(A), the density of abundant numbers, namely 0.247619608 < d(A) < 0.247619658. We also show the count of primitive covering numbers up to x is O( xexp((-12√log 2 + ε)√log x log log x)), which is substantially smaller than the corresponding bound for primitive abundant numbers.