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), 5358 Request permission
Abstract:
Let ${\Delta _k}(x)$ and ${\Delta _k}β (x)$ 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: \[ {\Delta _k}β (x)  x{\Delta _k}(x) = O({x^{1 + 3/(4k + 1) + \varepsilon }})\] and \[ \frac {1}{x}\int _1^x {{\Delta _k}(t)dt = O({x^{3/(4k + 1)\varepsilon }}),} \] where $\varepsilon > 0$.References

A. Axer, Γber einige Grenzertsatze, S.B. Akad. Wiss. Wien Ha 120 (1911), 12531298.
 E. C. Titchmarsh, The Theory of the Riemann ZetaFunction, Oxford, at the Clarendon Press, 1951. MR 0046485
 A. M. Vaidya, On the order of the error function of the squarefree numbers, Proc. Nat. Inst. Sci. India Part A 32 (1966), 196β201. MR 249378
 Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Mathematische Forschungsberichte, XV, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963 (German). MR 0220685
Additional Information
 © Copyright 1971 American Mathematical Society
 Journal: Proc. Amer. Math. Soc. 28 (1971), 5358
 MSC: Primary 10.42
 DOI: https://doi.org/10.1090/S0002993919710271044X
 MathSciNet review: 0271044