  Let $R_k(x)$ denote the error incurred by approximating the number of
$k$-free integers less than $x$ by $x/\zeta(k)$. It is well known that
$R_k(x)=\Omega(x^{\frac{1}{2k}})$, and widely conjectured that
$R_k(x)=O(x^{\frac{1}{2k}+\epsilon})$. By establishing weak linear independence
of some subsets of zeros of the Riemann zeta function, we establish an
effective proof of the lower bound, with significantly larger bounds on the
constant compared to those obtained in prior work. For example, we show that
$R_k(x)/x^{1/2k} > 3$ infinitely often and that $R_k(x)/x^{1/2k} < -3$
infinitely often, for $k=2$, $3$, $4$, and $5$. We also investigate $R_2(x)$
and $R_3(x)$ in detail and establish that our bounds far exceed the
oscillations exhibited by these functions over a long range: for
$0<x\leq10^{18}$ we show that $|R_2(x)| < 1.12543x^{1/4}$ and $|R_3(x)| <
1.27417x^{1/6}$. We also present some empirical results regarding gaps between
square-free numbers and between cube-free numbers.
