Is there an elementary proof that the Mertens function is not $O(x^\theta)$ if $\theta <1/2$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T09:51:46Z http://mathoverflow.net/feeds/question/47436 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47436/is-there-an-elementary-proof-that-the-mertens-function-is-not-ox-theta-if Is there an elementary proof that the Mertens function is not $O(x^\theta)$ if $\theta <1/2$? Alastair Irving 2010-11-26T15:24:15Z 2010-11-26T18:47:53Z <p>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 $\theta&lt;\frac 12$. </p> <p>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.</p> http://mathoverflow.net/questions/47436/is-there-an-elementary-proof-that-the-mertens-function-is-not-ox-theta-if/47449#47449 Answer by engelbrekt for Is there an elementary proof that the Mertens function is not $O(x^\theta)$ if $\theta <1/2$? engelbrekt 2010-11-26T18:47:53Z 2010-11-26T18:47:53Z <p>As far as I know, even an elementary proof that $M(x)$ is unbounded is not known.</p>