Generalized Smirnov Statistics and the Distribution of Prime Factors

(1.2) log2 pj ≤ αj + β (1 ≤ j ≤ ω(n)), where α ≥ 0 and log2 y denotes log log y. The distribution of integers satisfying (1.1) is important in the study of the distribution of divisors of integers (see [3]; Ch. 2 of [4]). We present here estimates for Nk(x;α, β) = #{n ≤ x : ω(n) = k, (1.1)}, Mk(x;α, β) = #{n ≤ x : ω(n) = k, (1.2)}. It is a relatively simple matter, at least heuristically, to reduce the estimation ofNk(x;α, β) and Mk(x;α, β) to the estimation of a certain probability connected to Kolmogorov-Smirnov statistics. Let us focus on the upper bound for Nk(x;α, β). If we suppose that pk ≥ x for some small c, then for each choice of (p1, . . . , pk−1), the number of possible pk is ≪ x/(p1 · · · pk−1 log x). Since ∑ p≤y 1/p ≈ log2 y, given a well-behaved function f , by partial summation we anticipate that