Skip to main content
added a top-level tag; https://meta.mathoverflow.net/questions/1457/why-are-mo-tags-formatted-as-they-are
Link
Martin Sleziak
  • 4.7k
  • 4
  • 35
  • 40
Notice removed Canonical answer required by Basj
Bounty Ended with kodlu's answer chosen by Basj
Notice added Canonical answer required by Basj
Bounty Started worth 50 reputation by Basj
Source Link
Basj
  • 587
  • 5
  • 15

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).