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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).

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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?

  • $\begingroup$ 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. $\endgroup$ – Greg Martin Jan 9 at 18:09
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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.

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    $\begingroup$ I guess c is the imaginary part of the first non trivial zero of zeta ? $\endgroup$ – Sylvain JULIEN Jan 9 at 17:49
  • $\begingroup$ 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. $\endgroup$ – Greg Martin Jan 9 at 18:12
  • $\begingroup$ @SylvainJULIEN, yes. Basj, I will see if I can track down a PDF. $\endgroup$ – kodlu Jan 9 at 21:24
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    $\begingroup$ @Basj, how can I get the PDF to you? I don't have a webpage I can upload to. $\endgroup$ – kodlu Jan 10 at 11:28
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    $\begingroup$ Teaching you to fish: :) Typing "Fiorilli" into The Mathematics Genealogy Project uncovers the title of his thesis, Irrégularités dans la distribution des nombres premiers et des suites plus générales dans les progressions arithmétiques; the first Google hit gets you the PDF. It definitely uses complex methods, specifically the explicit formula. $\endgroup$ – Greg Martin Jan 15 at 17:13

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