A rate estimate in Billingsley's theorem for the size distribution of large prime factors

Let {P-j(m) : 1 less than or equal to j less than or equal to omega(m)} den ote the decreasing sequence of distinct prime factors of a positive integer m. We provide an asymptotic expansion for the distribution function F-n(<(alpha)over right arrow>(k)) := v(n) {m: P-j(m) > n(alpha j)(1 less th an or equal to j less than or equal to k)} which is valid uniformly in a large range for ((alpha) over arrow (k)) := ( alpha(1), ... , alpha(k)). When k greater than or equal to 2, we give an as ymptotic formula for the same quantity which holds with no restriction at a ll on (<(alpha)over right arrow>(k)). A sample consequence of this second r esult is that, given any fixed k greater than or equal to 2, the formula v(n) {P-k(m) less than or equal to y} = r(k)(u) {1 + O(1/ log y)} holds uniformly for 2 less than or equal to y less than or equal to n, wher e u is defined by n = y(u) and r(k) is a suitable distribution function.