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 \[ \Delta _k^\ast (x) = O({x^{2/(2k + 1)}}\exp \{ A\log x{(\log \log x)^{ - 1}}\} ).\]References
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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