TITLE: Oscillation of the summatory Möbius function
QUESTION [9 upvotes]: 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.

REPLY [3 votes]: You may also find quantitative results concerning error term from prime number theorem in the article S. B. Stechkin, A. Yu. Popov, “The asymptotic distribution of prime numbers on the average”, Uspekhi Mat. Nauk, 51:6(312) (1996), 21–88, english translation is published in Russ. Math. Surv. 51 (1996), pp. 1025-1092.