Let $M(x) = \sum_{n\leq x} \mu(n)$ and $m(x) = \sum_{n\leq x} \frac{\mu(n)}{n}$, where $\mu(n)$ is the Möbius function. We know that (it is not the best known bounds): $$\limsup_{x \to \infty} M(x)x^{-1/2} > 1$$ $$\liminf_{x \to \infty} M(x)x^{-1/2} < -1$$

Are there any references showing that (if true) we also have for some $c>0$: $$\limsup_{x \to \infty} m(x)x^{1/2} > c$$ $$\liminf_{x \to \infty} m(x)x^{1/2} < -c$$