Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T11:26:38.797Z Has data issue: false hasContentIssue false
  • English
  • Français

Squarefree numbers on short intervals

Published online by Cambridge University Press:  26 February 2010

R. R. Hall
R. R. Hall
Affiliation:
Department of Mathematics, University of York, Heslington, York. YO1 5DD
Get access

Extract

It is a familiar fact that the sequence of K-free numbers, i.e. those having no K-th power divisor greater than 1, has asymptotic density 1/ζ(K). Let χ(K) denote the characteristic function of this sequence, χ(2) = χ. Roth [5] proved that, if k/(n3/13(log n)4/13) → ∞, then

Type
Research Article
Information
Mathematika , Volume 29 , Issue 1 , June 1982 , pp. 7 - 17
Copyright
Copyright © University College London 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bellman, R. and Shapiro, H. N.. The distribution of squarefree integers in small intervals. Duke Math. J., 21 (1954), 629–637.CrossRefGoogle Scholar
2.Carlitz, L.. On a problem in additive arithmetic II. Quarterly J. of Math. (Oxford), 3 (1932), 273–290.CrossRefGoogle Scholar
3.Halberstam, H. and Roth, K. F.. On the gaps between consecutive k-free integers. J. London Math. Soc., 26 (1951), 268–273.CrossRefGoogle Scholar
4.Richert, H.-E.. On the difference between consecutive squarefree numbers. J. London Math. Soc., 29 (1954), 16–20.CrossRefGoogle Scholar
5.Roth, K. F.. On the gaps between squarefree numbers. J. London Math. Soc, 26 (1951), 263–268.CrossRefGoogle Scholar
6.Titchmarsh, E. C.. The theory of the Riemann Zeta function (Oxford, 1951).Google Scholar