Lower bounds on zeta(s+it) for fixed s - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T06:08:20Z http://mathoverflow.net/feeds/question/23873 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23873/lower-bounds-on-zetasit-for-fixed-s Lower bounds on zeta(s+it) for fixed s Fedor Petrov 2010-05-07T16:27:30Z 2012-04-12T18:49:48Z <p>This is most probably widely known and discussed here many times, so I am preliminay sorry.</p> <p>Does Riemann conjecture imply some lower estimates on values, say $|\zeta(3/4+it)|$ for real $t$, when $|t|$ tends to infinity?</p> <p>Are any such results known without assuming Riemann conjecture (many doubts here)?</p> <p>Thanks!</p> http://mathoverflow.net/questions/23873/lower-bounds-on-zetasit-for-fixed-s/23892#23892 Answer by engelbrekt for Lower bounds on zeta(s+it) for fixed s engelbrekt 2010-05-07T20:14:45Z 2010-05-07T20:14:45Z <p>Yes, such conditional results are covered in Chapter 14 of the standard reference - the second edition of The Theory of the Riemann Zeta-Function by E. C. Titchmarsh. This edition has end-of-chapter notes by D. R. Heath-Brown bringing it up to date as of 1986.</p> <p>In particular a lower bound $$|\zeta(3/4 + it)| \gg e^{-c\sqrt{\log(t)}/\log\log(t)}$$ holds with some $c > 0$, conditionally on RH. See page 384 of the cited reference.</p> <p>Such results are only known unconditionally for a region to the left of the line $\sigma = 1$ that narrows to zero width as $t \rightarrow {\pm}\infty$. Not coincidentally, the best zero-free region known is also of this form. See page 135 of the cited reference for the best result of this kind.</p>