# 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).

## This question has an open bounty worth +50 reputation from Basj ending in 3 days.

The question is widely applicable to a large audience. A detailed canonical answer is required to address all the concerns.

This bounty is about the second part of the question: are there some elementary proofs (not using complex analysis directly) about the frequency of change of sign of the Mertens function?

• Regarding your second question: we can start by asking whether there is an elementary proof of classical oscillation theorems, for example that $\psi(x)-x$ changes sign infinitely often. – Greg Martin Jan 9 at 18:09

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).
• To the best of my knowledge, nothing stronger than $\Omega(\log y)$ is known for the number of sign changes of any of the usual number-theoretic functions and their error terms. Even showing that the number of sign changes grows faster than any constant multiple of $\log y$ seems quite difficult. Daniel Fiorilli pointed out in his thesis that one of the reasons we can't do better (despite the truth probably being around $\sqrt y$) is that these proofs use a many-times-averaged version of these functions, which actually do have only $O(\log y)$ sign changes. – Greg Martin Jan 9 at 18:12