I started to read this preprint: https://arxiv.org/abs/2010.03696

In it, the author states that $\sum_{n\leq x}\mu_{k}(n)=\zeta(k)^{-1}x+O(x^{1/k})$ and that under RH, the exponent in the error term becomes $\frac{1}{k+1}$ (where $\mu_{k}$ is the indicator of $k$-free numbers).

What would an exponent of the form $\frac{1}{\sqrt{k(k+1)}}$ imply towards RH? Conversely, assuming the supremum of the real parts of the non trivial zeros of the Riemann zeta function is $1-\varepsilon$ for some $\varepsilon >0$, what would it imply for the value of the considered exponent?