It is well known that the [Mertens function](https://en.wikipedia.org/wiki/Mertens_function)

$$M(x)=\sum _{n\leq x}\mu(n)$$

has infinitely many zeros, and this seems to be a [short proof](https://math.stackexchange.com/a/2345591/109021).

**Are there known results about *how often* the Mertens function is 0?** (i.e. how many times on average between $1$ and $x$)

Also, is it possible to prove that it vanishes infinitely often with elementary techniques only, and no complex analysis / Zeta function? (In the same way the PNT has been proved elementary by Selberg/Erdös around 1950).