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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the fractional parts of $n/j,$ $j=o(n)$
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by John Isbell and Stephen Schanuel PDF
Proc. Amer. Math. Soc. 60 (1976), 65-67 Request permission

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
    L. E. Dickson, History of the theory of numbers, vol. I, Stechert, New York, 1934.
  • Arnold Walfisz, Über Gitterpunkte in mehrdimensionalen Ellipsoiden, Math. Z. 35 (1932), no. 1, 212–229 (German). MR 1545298, DOI 10.1007/BF01186558
  • 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
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 60 (1976), 65-67
  • MSC: Primary 10H20; Secondary 10K99
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0429796-6
  • MathSciNet review: 0429796