The density of covering numbers

Abstract: In 1950, Erd\H{o}s introduced covering systems--finite collections of arithmetic progressions whose union contains every integer. They featured in some of his favorite problems, many of which are still open.  In 1979, answering one of Erd\H{o}s's questions, Haight introduced covering numbers: positive integers $n$ for which a covering system can be constructed with distinct moduli that are divisors of $n$. If no proper divisor of $n$ is a covering number, we call $n$ a primitive covering number.  We establish an upper bound on the number of primitive covering numbers, from which it follows that the set of covering numbers has a natural density. By refining techniques used to bound the density of abundant numbers, we obtain relatively tight bounds for the density of covering numbers and, in the process, improve the bounds on the density of abundant numbers as well.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)