## On the order of some error functions related to $k$-free integers

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- by V. S. Joshi
- Proc. Amer. Math. Soc.
**35**(1972), 325-332 - DOI: https://doi.org/10.1090/S0002-9939-1972-0337839-X
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Erratum: Proc. Amer. Math. Soc.

**51**(1975), 251-252.

## 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 not exceeding

*x*. We prove that on the assumption of Riemann hypothesis, we have \[ \Delta {’_k}(x) - x{\Delta _k}(x) = O({x^{1 + 1/2k + \varepsilon }})\] and \[ \frac {1}{x}\int _1^x {{\Delta _k}(t)dt = O({x^{1/2k + \varepsilon }}),} \] for arbitrary $\varepsilon > 0$.

## References

- E. Hille,
- D. Suryanarayana and R. Sitaramachandra Rao,
*On the order of the error function of the $k$-free integers*, Proc. Amer. Math. Soc.**28**(1971), 53–58. MR**271044**, DOI 10.1090/S0002-9939-1971-0271044-X - E. C. Titchmarsh,
*The Theory of the Riemann Zeta-Function*, Oxford, at the Clarendon Press, 1951. MR**0046485**
—, - 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**

*Analysis*. Vol. I, Blaisdell, Waltham, Mass., 1964. Theorem B.2.3, p. 333.

*The theory of functions*, Clarendon Press, Oxford, 1939, pp. 44-45.

## Bibliographic Information

- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**35**(1972), 325-332 - MSC: Primary 10H25
- DOI: https://doi.org/10.1090/S0002-9939-1972-0337839-X
- MathSciNet review: 0337839