Properties of Mertens Function

Question

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.

Answer 1

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.

Answer 2

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

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

3. 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 $$M(n) << \sqrt{n} exp((\log n)^{1/2} (\log \log n)^{14})$$