On the order of the error function fo the $k$-free integers

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 $\varepsilon > 0$.