On the number of positive integers ≦ x and free of prime factors > y

Let Ψ(x, y) denote the number of positive integers ≦ x and free of prime factors > y. De Bruijn showed that the relation Ψ(x, x 1 u ) ∼ xϱ(u) holds, as x → ∞, uniformly in the range 1 ≦ u ≦ ( log x) 3 8 − ϵ . We extend this range to 1 ≦ u ≦ log x ( log log x) 5 3 + ϵ , and give a similar, but weaker estimate for the range 1 ≦ u ≦ log x (1 + ϵ) log log x . We also prove a short interval result.