Updating an upper bound of Erik Westzynthius

Abstract

Inspired by a paper of Erik Westzynthius,we build on work of Harlan Stevens and Hans-Joachim Kanold. Let $k \gt 2$ be the number of distinct prime divisors of a positive integer $n$. In 1977, Stevens used Bonferroni inequalities to get an explicit upper bound on Jacobsthal's function $g(n)$, which is related to the size of largest interval of consecutive integers none of which are coprime to $n$. Letting $u(k)$ be the base $2$ $\log$ of this bound, Stevens showed $u(k)$ is $O((\log k)^2)$, improving upon Kanold's exponent $O(\sqrt{k})$. We use elementary methods similar to those of Stevens to get $u(k)$ is $O(\log k(\log\log k))$ in one form and $O(\sigma^{-1}(n)\log k)$ in another form. We also show how these bounds can be improved for small $k$.