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Proof of a conjecture about the distribution of divisors of integers in residue classes

Published online by Cambridge University Press:  24 October 2008

P. Erdös  and
R. R. Hall
P. Erdös
Affiliation:
Trinity College, Cambrigde, and University of York
R. R. Hall
Affiliation:
Trinity College, Cambrigde, and University of York
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Extract

Let k be a positive integer and F(x, k) denote the number of integers n < x which have a divisor in every residue class prime to k. Erdös (1) proved that for every fixed ε > 0, we have F(x, k) ˜ x when

and conjectured the following result, which we prove in this paper.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 79 , Issue 2 , March 1976 , pp. 281 - 287
Copyright
Copyright © Cambridge Philosophical Society 1976

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References

REFERENCES

(1)Erdös, P.On the distribution of divisors of integers in residue classes (mod d). Bull. Soc. Math. Grèce 6 Fasc 1 (1965), 27–36.Google Scholar
(2)Hall, R. R.A conjecture of Erdös in number theory. Acta Arithmetica (to appear).Google Scholar
(3)Erdös, P. and Rényi, A.Probabilistic methods in group theory. Journal Analyse Math. 14 (1965), 127–38.Google Scholar