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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.

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Copyright information

© Birkhäuser Boston 1987

Authors and Affiliations

  • Douglas Hensley
    • 1
  1. 1.Texas A&M UniversityCollege StationUSA

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