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Square-full numbers in short intervals

Published online by Cambridge University Press:  24 October 2008

D. R. Heath-Brown
D. R. Heath-Brown
Affiliation:
Magdalen College, Oxford
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Extract

A positive integer n is called square-full if p2|n for every prime factor p of n. Let Q(x) denote the number of square-full integers up to x. It was shown by Bateman and Grosswald [1] that

Bateman and Grosswald also remarked that any improvement in the exponent would imply a ‘quasi-Riemann Hypothesis’ of the type for . Thus (1) is essentially as sharp as one can hope for at present. From (1) it follows that, for the number of square-full integers in a short interval, we have

when and y = o (x½). (It seems more suggestive) to write the interval as (x, x + x½y]) than (x, x + y], since only intervals of length x½ or more can be of relevance here.)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 1 , July 1991 , pp. 1 - 3
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Bateman, P. T. and Grosswald, E.. On a theorem of Erdös and Szekeres. Illinois J. Math. 2 (1958), 88–98.CrossRefGoogle Scholar
[2]Jia, C-H.. The square-full integers in the short interval. Acta Math. Sinica 5 (1987), 614–621.Google Scholar
[3]Liu, H.. On square-full numbers in short intervals. Acta Math. Sinica 6 (1990), 148–164.Google Scholar
[4]Richert, H.-E.. Über die Anzahl Abelscher Gruppen gegebener Ordnung. I. Math. Z. 56 (1952), 21–32.CrossRefGoogle Scholar
[5]Shiu, P.. On square-full integers in a short interval. Glasgow Math. J. 25 (1984), 127–134.CrossRefGoogle Scholar