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On the Distribution of Square-Free Numbers

Published online by Cambridge University Press:  20 November 2018

C. Hooley
C. Hooley*
Affiliation:
University College, Cardiff, Cardiff, Wales
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Erdös [1] has shown that, if the square-free numbers in ascending order be denoted by s1, s2, … , sn, … , then for 0 ≦ γ ≦ 2

as x → ∞. In this paper we shall extend this result by proving that the asymptotic formula in fact holds for the wider range 0 ≦ γ ≦ 3.

Similar results have been obtained previously by the author in respect of both the sequence of numbers expressible as the sum of two squares and also sequences of numbers relatively prime to given large integers, although the method used here differs from that of the earlier papers [2; 3].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 6 , December 1973 , pp. 1216 - 1223
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Erdös, P., Some problems and results in elementary number theory, Publ. Math. Debrecen 2 (1951), 103–109.Google Scholar
2. Hooley, C., On the difference of consecutive numbers prime to n, I, Acta Arith. 8 (1963), 343–347. 3# On the intervals between numbers that are sums of two squares, Acta Math. 127 (1971), 279-297.Google Scholar
4. Mirsky, L., Arithmetical pattern problems relating to divisibility by r-th powers, Proc. London Math. Soc. 50 (1949), 497–508.Google Scholar
5. Richert, H. E., On the difference between consecutive square-free numbers, J. London Math. Soc. 29 (1954), 16–20.Google Scholar
6. Selberg, A., On the normal density of primes in small intervals and the difference between consecutive primes, Archiv Math, og Naturvid. 47 (1943), 87–106.Google Scholar