In fact many texts give *equivalent* statements to RH in terms of $M(N)$ and related arithmetic functions. In particular RH is equivalent to the following: for any $\epsilon>0$, there is $N_0$ depending on $\epsilon$, such that for any $N>N_0$ we have $|M(N)|<N^{1/2+\epsilon}$.

Informally, this means that if RH fails, then $|M(N)|$ is occasionally much larger than $\sqrt{N}$, and vice versa. It is a subtle question how large $|M(N)|$ can really get if RH holds. For the state-of-the-art in this direction and for what more can be expected, see [Soundararajan's paper][1].


  [1]: http://arxiv.org/abs/0705.0723