Let the Mertens function $$M(x) = \sum_{n \le x} \mu(n)$$ I assume (perhaps foolishly) that it is known that $M(x)$ changes sign infinitely often. If that's true, the question is a quantitative version :

How many sign changes of $M(x)$ are there between $1$ and $y$ (asymptotically) ?

**ADDITION* GH from MO cites a result which gives a logarithmic number of changes. This, while better than nothing, is not (empirically the truth): for $N=1000000,$ you get around $5500$ sign changes, for $N=10000000,$ around $12000,$ and here is the graph of the total number of sign changes.This looks square-rootish. Now, what is even more interesting is that for a symmetric random walk, the number of returns to the origin is asymptotic to $\frac{2}{\pi} \sqrt{N},$ which is much smaller than this. (see https://math.stackexchange.com/questions/1338097/expected-number-of-times-random-walk-crosses-0-line for deviation). The difference is even more striking, if you remember that a lot of numbers are non-square-free.