TITLE: Is there an elementary proof that the Mertens function is not $O(x^\theta)$ if $\theta <1/2$?
QUESTION [15 upvotes]: The Mertens function is the partial sums of the Moebius function:
$M(x)=\sum_{n\leq x}\mu(n)$
Since the zeta-function has a zero on the critical line it follows that $M(x)\ne O(x^\theta)$ for any $\theta<\frac 12$.  
Does anyone know if there is an elementary proof of this statement?  (By elementary I mean a proof which does not depend on complex analysis, in particular the existance of a zero of $\zeta$).  even an elementary proof of $M(x)$ being unbounded would be interesting to me.

REPLY [4 votes]: As far as I know, even an elementary proof that $M(x)$ is unbounded is not known.