The distribution of $k$-free numbers and the derivative of the Riemann zeta-function

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 $M_k(x)={\sum }_{n\le x}{\mu }_k(n)-\frac{x}{\zeta (k)}$ 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.