User juan arias de reyna - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T11:49:12Z http://mathoverflow.net/feeds/user/17228 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73098/negative-values-of-riemann-zeta-function-on-the-critical-line/73143#73143 Answer by Juan Arias de Reyna for Negative values of Riemann zeta function on the critical line. Juan Arias de Reyna 2011-08-18T12:38:41Z 2011-08-18T13:49:32Z <p>The zeta function is real on the critical line only at the zeros and at Gram points, this is because zeta(1/2+it)=exp(-ivartheta(t)) Z(t). </p> <p>At the Gram point g_k we have by definition vartheta(g_k)=pi k. so that zeta(1/2+ig_k) =(-1)^k Z(g_k).</p> <p>Now a Gram point g_k is said a good Gram point if (-1)^k Z(g_k) >0. In other case it is said a bad Gram point.<br> Since it appear improbable a zero just at a Gram point. You are asking if there exists bad Gram points, there are plenty. The first few bad Gram points are g_126, g_134, g_195, g_211, ...</p> <p>g_126 = 282.45472082346217461077</p> <p>In fact it is proved there are infinite bad Gram points. </p> <p>Also we may easily obtain large negative values. For example using data of T. Kotnik "Computational estimation of the order of zeta(1/2+it) Math of Comp. (2003) we easily locate the point t = grampoint(2601005843707) were we have</p> <p>zeta(0.5+i t) = -119.6304321077241661374</p> <p>This is easily confirmed in mpmath (or Mathematica) ( grampoint(2601005843707) = 669980906189.53552206792 ).</p>