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 n for which there exists a distinct covering system where every modulus divides n n . We establish that the set C \mathcal {C} of covering numbers possesses a natural density d ( C ) d(\mathcal {C}) and prove that 0.103230 > d ( C ) > 0.103398 0.103230 > d(\mathcal {C}) > 0.103398 . Our approach adapts methods developed by Behrend [Akad. Berlin (1932), pp. 322–328; Akad. Berlin (1932), pp. 280–293] and Deléglise [Experiment. Math. 7 (1998), pp. 137–143] for bounding the density of abundant numbers, by introducing a function c ( n ) c(n) that measures how close an integer n n is to being a covering number with the property that c ( n ) ≤ h ( n ) = σ ( n ) / n c(n) \leq h(n) = \sigma (n)/n . However, computing d ( C ) 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 ( A ) d(\mathcal {A}) , the density of abundant numbers, namely 0.247619608 > d