The Distribution of Ω(n) among Numbers with No Large Prime Factors

Abstract

The main result concerns the distribution of Ω(n) within

$$\text{S(x,y)} = \left\{ {\text{n}:1 \leqslant \text{n} \leqslant \text{x}\,\text{and}\,\text{p} \leqslant \text{y}\,\text{if}\,\text{p}\left| \text{n} \right.} \right\}.$$

There is an average value k₀ for Ω(n), and a dispersion parameter V,such that for k not too far from k₀, and for large x, y with

$$2\,\,\log \log \,\text{x}\,\text{ + }\,\text{1} \leqslant \text{log}\,\,\text{y} \leqslant (\log \text{x})^{3/4} .$$

the number of solutions n of Ω(n) = k in S(x,y) is roughly exp(-V(k-k₀)²) times the number of solutions n of Ω(n) = k₀ in S(x,y).

In the course of the proof, machinery is developed which permits a sharpening in the same range of previous estimates for the local behaviour of ψ(x,y) as a function of x.