  Under the Riemann Hypothesis, we connect the distribution of $k$-free numbers
with the derivative of the Riemann zeta-function at nontrivial zeros of
$\zeta(s)$. Moreover, with additional assumptions, we prove the existence of a
limiting distribution of $e^{-\frac{y}{2k}}M_k(e^y)$ and study the tail of the
limiting distribution, where $M_k(x)=\sum_{n\leq x}\mu_k(n)-\frac{x}{\zeta(k)}$
and $\mu_k(n)$ is the characteristic function of $k$-free numbers. Finally, we
make a conjecture about the maximum order of $M_k(x)$ by heuristic analysis on
the tail of the limiting distribution.
