Let s1, s2, … denote the squarefree numbers in ascending order. In , Erdős showed that, if 0 ≤ γ ≤ 2, then
where B(γ) is a function only of γ. In 1973 Hooley  improved the range of validity of this result to 0 ≤ γ ≤ 3, and then later gained a further slight improvement by a method he outlined at the International Number Theory Symposium at Stillwater, Oklahoma in 1984. We have, however, independently obtained the better improvement that (1) holds for
in contrast to the range
derived by Hooley. The main purpose of this paper is to substantiate our new result. Professor Hooley has informed me that there are similarities between our methods as well as significant differences.