A property of the counting function of integers with no large prime factors

Abstract

For logy log log x → ∞ as x → ∞, ψ( Cx, y) ≈ Cψ( x, y) uniformly for C in compact subsets of (0, 1]. The condition logy log log x → ∞ is necessary. For fixed sufficiently large K, x → ∞, and y = exp( K log log x), we have for 1 2 ≤ C ≤ 1 that Ψ(Cx,y)≈C 1−1/KΨ(x,y) .