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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 order of the error function fo the $k$-free integers
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by D. Suryanarayana and R. Sitaramachandra Rao PDF
Proc. Amer. Math. Soc. 28 (1971), 53-58 Request permission

Abstract:

Let ${\Delta _k}(x)$ and ${\Delta _k}’ (x)$ be the error functions in the asymptotic formulae for the number and the sum of $k$-free integers $\leqq x$. On the assumption of the Riemann hypothesis, we prove the following results by elementary methods: \[ {\Delta _k}’ (x) - x{\Delta _k}(x) = O({x^{1 + 3/(4k + 1) + \varepsilon }})\] and \[ \frac {1}{x}\int _1^x {{\Delta _k}(t)dt = O({x^{3/(4k + 1)\varepsilon }}),} \] where $\varepsilon > 0$.
References
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  • A. M. Vaidya, On the order of the error function of the square-free numbers, Proc. Nat. Inst. Sci. India Part A 32 (1966), 196–201. MR 249378
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 28 (1971), 53-58
  • MSC: Primary 10.42
  • DOI: https://doi.org/10.1090/S0002-9939-1971-0271044-X
  • MathSciNet review: 0271044