# nt.number theory – Error term for the summatory function of \$k\$-free numbers indicator and RH

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

In it, the author states that $$sum_{nleq x}mu_{k}(n)=zeta(k)^{-1}x+O(x^{1/k})$$ 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?