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

• Douglas Hensley
Chapter
Part of the Progress in Mathematics book series (PM, volume 70)

## 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.

## Keywords

Error Term Prime Divisor Prime Number Theorem Large Prime Factor Probabilistic Number Theory
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
K. Alladi, The Turan-Kubilius inequality for integers without large prime factors, J. Fur die reine u. angew. Math. 335 (1982) 180–196.
2. 2.
K. Alladi, An Erdös-Kac theorem for integers without large prime factors, Acta Arith. (to appear).Google Scholar
3. 3.
P.D.T.A. Elliot, Probabilistic Numer Theory I, Grundlehren der mathematichen wissenchaften 239, springer verlag, NY 1989 (p. 74).Google Scholar
4. 4.
D. Hensley, A property of the counting function of integers with no large prime factors, J. of Number Th. 22, (1986), 46–74.
5. 5.
A. Hildebrand, On the local behavior of ψ(x,y), Trans. Am. Math. Soc,. (1986), to appear.Google Scholar
6. 6.
K. Prachar, Primzahlverteilung, Grundlehren der mathematischen Wissenschaften 41, Berlin 1957.Google Scholar
7. 7.
A. Seiberg, On the normal density of primes in small intervals and the difference between consecutive primes. Archk. Math. Naturvid 47, No. 6 (1943) 87–105.Google Scholar