Is it possible to reconstruct $\zeta$-function knowing its zeroes? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T20:34:51Z http://mathoverflow.net/feeds/question/102338 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102338/is-it-possible-to-reconstruct-zeta-function-knowing-its-zeroes Is it possible to reconstruct $\zeta$-function knowing its zeroes? Math-player 2012-07-16T10:07:35Z 2012-07-16T17:07:39Z <p>Hello, Is it possible to reconstruct the Riemann zeta function given the precise location of its infinitely many zeroes? Thanks</p> http://mathoverflow.net/questions/102338/is-it-possible-to-reconstruct-zeta-function-knowing-its-zeroes/102342#102342 Answer by juan for Is it possible to reconstruct $\zeta$-function knowing its zeroes? juan 2012-07-16T10:29:07Z 2012-07-16T10:29:07Z <p>This is well known (Riemann could have writen it)</p> <p><code>$$\zeta(s)=\frac{1}{2}\frac{\pi^{s/2}}{(s-1)\Gamma(1+s/2)}\prod_{\Im\rho &gt; 0}\Bigl\{ \Bigl(1-\frac{s}{\rho}\Bigr)\Bigl(1-\frac{s}{\overline{\rho}}\Bigr)\Bigr\}$$</code></p> <p>Here $\rho$ runs through the non trivial zeros with positive imaginary part.</p> <p>It is this what you call reconstruct?</p>