Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the fractional parts of $ n/j,$ $ j=o(n)$


Authors: John Isbell and Stephen Schanuel
Journal: Proc. Amer. Math. Soc. 60 (1976), 65-67
MSC: Primary 10H20; Secondary 10K99
MathSciNet review: 0429796
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Abstract: Dirichlet's result that if $ J(n) = o(n)$ but $ {n^{1/2}} = o(J(n))$, the numbers $ n/j$ for $ j = 1, \ldots ,J(n)$ are nearly uniformly distributed modulo 1 (with error $ \to 0$ as $ n \to \infty $) is extended, $ {n^{1/2}}$ being replaced by $ {n^\alpha }$ for any $ \alpha > 0$.


References [Enhancements On Off] (What's this?)

  • [1] L. E. Dickson, History of the theory of numbers, vol. I, Stechert, New York, 1934.
  • [2] Arnold Walfisz, Über Gitterpunkte in mehrdimensionalen Ellipsoiden, Math. Z. 35 (1932), no. 1, 212–229 (German). MR 1545298, 10.1007/BF01186558
  • [3] Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Mathematische Forschungsberichte, XV, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963 (German). MR 0220685

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0429796-6
Keywords: Uniform distribution
Article copyright: © Copyright 1976 American Mathematical Society