lower bound for $\Re\zeta(1+it)$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T21:42:41Z http://mathoverflow.net/feeds/question/80973 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80973/lower-bound-for-re-zeta1it lower bound for $\Re\zeta(1+it)$ asd 2011-11-15T11:35:03Z 2011-11-15T12:38:35Z <p>Hi</p> <p>is there any lower bound for $\Re\zeta(1+it)$. </p> <p>I did try with computer until some ordinate and I saw $\Re\zeta(1+it)>0$.</p> <p>If it is true, is there any reference to prove it. thanks</p> http://mathoverflow.net/questions/80973/lower-bound-for-re-zeta1it/80976#80976 Answer by quid for lower bound for $\Re\zeta(1+it)$ quid 2011-11-15T12:23:46Z 2011-11-15T12:33:09Z <p>No, this is not true; see Table 5 in <a href="http://arxiv.org/abs/1001.2962" rel="nofollow">http://arxiv.org/abs/1001.2962</a> and conclusions. In particular the real part is negative for $t=682112.9$ ; and this the smallest value given there (and it was found via testing at steps of size $.1$ so perhaps no much smaller ones were missed). </p> <p>You might also be interested <a href="http://mathoverflow.net/questions/73098" rel="nofollow">in this question's</a> answers for related information for the critical line $1/2 + it$</p> http://mathoverflow.net/questions/80973/lower-bound-for-re-zeta1it/80977#80977 Answer by Johan Andersson for lower bound for $\Re\zeta(1+it)$ Johan Andersson 2011-11-15T12:29:01Z 2011-11-15T12:38:35Z <p>There is no lower bound for Re$(\zeta(1+it))$. This can be found in Lamzouri's paper <a href="http://www.math.uiuc.edu/~lamzouri/distribzeta.pdf" rel="nofollow">http://www.math.uiuc.edu/~lamzouri/distribzeta.pdf</a> (see e.g. page 3, formula (5)). (Choose $\tau$ large enough, $\theta_1=3\pi/4$ and $\theta_2=5\pi/4$ for and substract). The results of Lamzouri however also implies that on average the argument of $\zeta(1+it)$ is small and that Re$(\zeta(1+it))$ is positive more often than it is negative.</p>