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

Joshi, V. S.

doi:10.1090/S0002-9939-1972-0337839-X

On the order of some error functions related to $k$-free integers
HTML articles powered by AMS MathViewer

by V. S. Joshi PDF

Proc. Amer. Math. Soc. 35 (1972), 325-332 Request permission

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

Analysis

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

The theory of functions

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