I am trying to fully understand the implications of $M(n)=O\left(n^{\frac{1}{2}+\epsilon}\right)$, where $M(n)$ is Mertens function, being equivalent to Riemann Hypothesis.
(i) Is the equivalence biconditional? Why?
(ii) What is the meaning of $\epsilon$ and how big can it be? For instance, say $M(n)=O\left(\sqrt{n} \cdot \log\log(n)^{\theta}\right)$, where $\theta$ is unbounded. Is this still equivalent to $M(n)=O\left(n^{\frac{1}{2}+\epsilon}\right)$? And what if $\left |M(n)\right | \sim \frac{n^{\frac{3}{4}}}{2\log(n)}$, where we have that $\frac{n^{\frac{3}{4}}}{2\log(n)}$ grows slightly faster than $\sqrt{n}\cdot \log\log(n)$, but $\lim_{n\to\infty} \frac{\frac{n^{\frac{3}{4}}}{2\log(n)}}{\sqrt{n}\cdot \log\log(n)} = \infty$? Is this still equivalent to $M(n)=O\left(n^{\frac{1}{2}+\epsilon}\right)$?
Thanks for your help!