Properties of Mertens Function I am astounded by how little information about Mertens function M(n) (partial sums of the Möbius function) is on the Internet. Thus, I would be thankful if someone could clear up some of my confusion. 
First, I learned that PNT (prime number theorem) $\iff M(n)/n \rightarrow 0$  as $n\rightarrow \infty$
This makes sense as M(n) is the count of square-free integers up to n that have an even number of prime factors, minus the count of those that have an odd number, and I would expect these to cancel out in their contribution to the quotient as $n \rightarrow \infty$.
If |M(n)| is bounded by B, couldn't we conclude $M(n) = O(B)$? If not, then is M(n) finite for all n but unbounded? I know $M(n)<n<\infty$ for all n.
Furthermore, does anyone happen to know the best big O for M(n)? Does anyone know any online sources that exposit on M(n)?
I am thankful to anyone that can provide some information.
 A: Sorry for the late answer on this, but the OP asked to the best big $O$ bound available, and it seems to be unanswered.  I thought it might be useful to have here for future reference.
The best unconditional big $O$ bound that I am aware of is originally due to Walfisz, but since his book is out of print, it is best to go to Ivic's book, Theorem 12.7, for the statement

There is an absolute constant $C>0$ such that $$M(x) = \sum_{n \leq x}
 \mu(n) \ll x \exp(-C \log^{3/5}x(\log \log x)^{-1/5}).$$

and proof.
In addition, some explicit bounds can be found here.
A: *

*$n^{1/2} \to \infty$ as $n\to \infty$, is there any problem with that?

*You can conclude that $M(n) = O(1)$ (or $O(B)$ if you want), if your $B$ is the same for all $n$.

*Getting information about $M(n)$ amounts to knowing zero-free region of $\zeta(s)$ by Perron's formula, so you would want to look up zero-free region results of $\zeta(s)$. Assume Riemann Hypothesis though, Soundararajan proved that $$M(n) << \sqrt{n} exp((\log n)^{1/2} (\log \log n)^{14})$$
A: I did a Google search for "Mertens" and got these:
http://mathworld.wolfram.com/MertensFunction.html
http://mathworld.wolfram.com/MertensConjecture.html
These seem like a good start.
