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Short intervals containing numbers without large prime factors

Published online by Cambridge University Press:  24 October 2008

Glyn Harman
Glyn Harman
Affiliation:
School of Mathematics, University of Wales College of Cardiff, Senghenydd Road, Cardiff CF2 4AG
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Extract

We denote, as usual, the number of integers not exceeding x having no prime factors greater than y by Ψ(x, y). We also write

The function Ψ(x, y) is of great interest in number theory and has been studied by many researchers (see [3], [5] and [6] for example). The function Ψ(x, z, y) has also received some attention (see [2], [4–6]). In this paper we shall try to obtain a positive lower bound for Ψ(x, z, y) with y as small as possible when z is about x½ in magnitude. We note that the approach in [5] and [6] allows y to be much smaller than is permissible here, but requires x/z to be smaller than any power of x in [6] (unless some conjecture like the Riemann Hypothesis is assumed), or needs in [5]. The following result was obtained by Balog[1].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 1 , January 1991 , pp. 1 - 5
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

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