The distribution of integers with at least two divisors in a short interval

Whereas, in usual cases, sieving by a set of primes may be fairly well controlled, through Buchstab’s identity, sieving by a set of integers is a much more complicated task. However, some fairly precise results are known in the case where the set of integers is an interval. We refer to the recent work [1] of the first author for specific statements and references. Define τ(n; y, z) := |{d|n : y < d 6 z}|, H(x, y, z) := |{n 6 x : τ(n; y, z) > 1}|, Hr(x, y, z) := |{n 6 x : τ(n; y, z) = r}|, H 2 (x, y, z) := |{n 6 x : τ(n; y, z) > 2}| = ∑