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Proceedings of the American Mathematical Society

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

 

 

Distribution of semi-$ k$-free integers


Authors: D. Suryanarayana and R. Sitaramachandra Rao
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
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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

$\displaystyle \Delta _k^\ast(x) = Q_k^\ast(x) - \alpha _k^\ast x = O({x^{1/k}}\exp \{ - A{\log ^{3/5}}x{(\log \log x)^{ - 1/5}}\} ),$

where A is an absolute positive constant. Further, on the assumption of the Riemann hypothesis, we prove that

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


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DOI: https://doi.org/10.1090/S0002-9939-1973-0311599-1
Keywords: Semi-k-free integers, Riemann zeta function, Riemann hypothesis
Article copyright: © Copyright 1973 American Mathematical Society