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).



How often does the Mertens function vanish?
Quoting an answer to the question [https://mathoverflow.net/questions/273845/oscillation-of-the-summatory-möbius-function](https://mathoverflow.net/questions/273845/oscillation-of-the-summatory-m%C3%B6bius-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.


