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On an inequality in the elementary theory of numbers

Published online by Cambridge University Press:  24 October 2008

H. A. Heilbronn
H. A. Heilbronn
Affiliation:
Trinity College
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Extract

Let a1, a2, …, an be a set of n positive integers. Then it is easily seen that the set of positive integers not divisible by any aν has a density, i.e. that if Nn(z) is the number of such integers not exceeding z, then z−1Nn(z) tends to a limit when z → ∞; and that

where

and where [u1, …, uμ] denotes the least positive common multiple of the positive integers u1, …, uμ.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 33 , Issue 2 , April 1937 , pp. 207 - 209
Copyright
Copyright © Cambridge Philosophical Society 1937

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References

* If f is the smallest positive integer satisfying the congruence b′ ≡ 1 (mod m), then f is called the order of b (mod m).

* Math. Ann. 109 (1934), 668–78.CrossRefGoogle Scholar An upper bound for the sum was obtained by Landau, , Acta Arithm. 1 (1935), 43–62CrossRefGoogle Scholar and by Erdös, and Turan, , Bull. de l'inst. de math. et méc. à l'univ. Koulycheff de Tomsk, 1 (1935), 144–7.Google Scholar