I am astounded by how little information about Mertens function M(n) (partial sums of the Mobius 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 but unbounded? It can never be infinite because $M(n)<n<\infty$ for all n.
Furthermore, does anyone happen to know the best big O M(n) is? Does anyone know any online sources that exposits on M(n)?
I am thankful to anyone that can provide some information.