most recent 30 from http://mathoverflow.net 2013-05-25T01:47:55Z http://mathoverflow.net/feeds/user/3074 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11074/partial-sums-of-multiplicative-functions/11174#11174 mnf 2010-01-08T19:43:03Z 2010-01-08T19:43:03Z

<p>Here is a largely historical remark: </p> <p>Littlewood showed that there exists $x$ so that $\pi(x)$ is greater than the log-integral $\mathrm{li}(x)$. In fact, $\pi(x) - \mathrm{li}(x)$ is (more or less) a linear combination of factors $x^{1/2+it}$, where $1/2+it$ are zeroes of $\zeta$. Form the multiplicative convolution with a suitable smooth function: you can thereby make the sum over only finitely many zeroes. Now find (by Dirichlet) some $x$ for which all the numbers $i t \log(x)$ are near multiples of $2 \pi$, etc. </p> <p>This shows that $\pi(x) - \mathrm{li}(x)$ is not bounded by $C \sqrt{x}$ for any $C$; in the case of Mobius, you get some $C$ but not any $C$, because it's harder to understand the coefficients of $x^{1/2+it}$. </p> <p>Finally, there are some results valid for any completely multiplicative function: See the paper by A. Granville and K. Soundararajan entitled "The spectrum of multiplicative functions." </p>