It is well known that the Mertens function

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

has infinitely many zeros, and this seems to be a short proof.

**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).