How often does the Mertens function vanish? 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).
 A: Quoting an answer to the question https://mathoverflow.net/questions/273845/oscillation-of-the-summatory-möbius-function

Let  $c=14.1347251…$. Then there are at least $(c/\pi-o(1))\log y$ sign changes in $M(x)$ in the interval $[1,y]$. This was proved by Kaczorowski and Pintz (Acta Math. Hungar. 48 (1986), 173-185, doi 10.1007/BF01949062).

This may well be the state of the art, but any comments on further results would of course be welcome. If no more is known then this question is a duplicate of the linked question.
A: Edit: Wrong, kept for the comments below.
I'm trying an answer with an elementary argument:
Let's assume there exists $A > 0$ such that $M(x) \neq 0$ for $x \geq A$.
We can assume that $M(x) \geq 1$ for $x \geq A$, as the proof would be the same in the negative case. Let:
$$K = \sum_{k \leq A} |M(k)|.$$
Then the well-known identity
$$1 = \sum_{d \leq x} M(x/d)$$
gives 
$$1 = \left|\sum_{d \leq x/A} M(x/d) + \sum_{x/A < d\leq x} M(x/d)\right|\geq \sum_{d \leq x/A} M(x/d)-K \geq \lfloor{x/A}\rfloor -K \xrightarrow[x\to\infty]\, +\infty$$
which is a contradiction.
As a conclusion, the Mertens function $M(x)$ has infinitely many zeros.
I wonder if a similar (but more clever) elementary argument can lead to an estimation of how often the Mertens function vanishes between $1$ and $y$.
