Contour integration of $\zeta(s)\zeta(2s)$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T15:36:24Z http://mathoverflow.net/feeds/question/58601 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58601/contour-integration-of-zetas-zeta2s Contour integration of $\zeta(s)\zeta(2s)$ unknown (google) 2011-03-16T02:59:32Z 2011-03-17T17:20:03Z <p>I have been looking at this for days and I am going insane. </p> <p>I need to show that for a dirichlet series equal to $\zeta(s)\zeta(2s)$ the sum of the coefficients less that x is $x\zeta(2)+O(x^(3/4))$ and then expand that to the $\Pi \zeta(ks)$ for all k in an effort to find the formula for the number of non-isomorphic abelian groups. </p> <p>I know that using perron's formula there is a simple pole at $s=1$ that gives a residue of $X\zeta(2)$ but I can't find a contour that converges or the exact error term. </p> http://mathoverflow.net/questions/58601/contour-integration-of-zetas-zeta2s/58604#58604 Answer by David Hansen for Contour integration of $\zeta(s)\zeta(2s)$ David Hansen 2011-03-16T03:10:38Z 2011-03-16T03:10:38Z <p>This case, at least, you can do by hand; if $c(n)$ is the $n$th coefficient of $\zeta(s)\zeta(2s)$, then</p> <p>$\sum_{n\leq X}c(n)=\sum_{nm^2\leq X}1=\sum_{m}\sum_{n\leq X/m^2}1 = \sum_{m &lt; \sqrt{X}}(m^{-2} X+O(1))=\zeta(2)X+O(\sqrt{X}).$</p> http://mathoverflow.net/questions/58601/contour-integration-of-zetas-zeta2s/58629#58629 Answer by Daniel Loughran for Contour integration of $\zeta(s)\zeta(2s)$ Daniel Loughran 2011-03-16T10:37:08Z 2011-03-17T03:59:36Z <p>If you havn't done so already, you might find it useful to look at the proof of theorem 12.2 on the divisor problem in Titchmarsh - The theory of the Riemann zeta function. Here, he goes through a detailed application of Perron's formula for the function $\zeta^k(s)$, which I believe to be very similar to your case.</p> <p>Indeed, for $s=\sigma + it$ and $\sigma>1/2$, $\zeta(2s)$ is absolutley convergent and hence uniformly bounded with respect to $t$. So this will not contribute to the contours that you choose (as long as $\sigma>1/2$!). What you need then is good upper bounds for the order of the zeta function in the critical strip. </p> <p>To get these, one normally finds the order of the function at two points, and then uses the <a href="http://en.wikipedia.org/wiki/Lindel%C3%B6f%27s_theorem" rel="nofollow">Phragmén–Lindelöf principle</a> for strips to get estimates for the function between these two points. For example, it is known that $\zeta(1/2 + it) = O(t^{1/4})$ (see <a href="http://en.wikipedia.org/wiki/Lindel%C3%B6f_hypothesis" rel="nofollow">The Lindelöf hypothesis</a>), although there are much better bounds available than that. This is all done in Titchmarsh's book.</p> <p>I hope this helps!</p>