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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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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Distribution of semi-$k$-free integers
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by D. Suryanarayana and R. Sitaramachandra Rao PDF
Proc. Amer. Math. Soc. 37 (1973), 340-346 Request permission

Abstract:

Let $Q_k^\ast (x)$ denote the number of semi-k-free integers $\leqq x$. It is known that $Q_k^\ast (x) = \alpha _k^\ast x + O({x^{1/k}})$, where $\alpha _j^\ast$ is a constant. In this paper we prove that \[ \Delta _k^\ast (x) = Q_k^\ast (x) - \alpha _k^\ast x = O({x^{1/k}}\exp \{ - A{\log ^{3/5}}x{(\log \log x)^{ - 1/5}}\} ),\] where A is an absolute positive constant. Further, on the assumption of the Riemann hypothesis, we prove that \[ \Delta _k^\ast (x) = O({x^{2/(2k + 1)}}\exp \{ A\log x{(\log \log x)^{ - 1}}\} ).\]
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 37 (1973), 340-346
  • MSC: Primary 10H25
  • DOI: https://doi.org/10.1090/S0002-9939-1973-0311599-1
  • MathSciNet review: 0311599