Lower bounds on zeta(s+it) for fixed s

2010-05-07T16:27:30Z

This is most probably widely known and discussed here many times, so I am preliminay sorry.

Does Riemann conjecture imply some lower estimates on values, say $|\zeta(3/4+it)|$ for real $t$, when $|t|$ tends to infinity?

Are any such results known without assuming Riemann conjecture (many doubts here)?

Thanks!

Answer by engelbrekt for Lower bounds on zeta(s+it) for fixed s

2010-05-07T20:14:45Z

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.

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.

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.

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Lower bounds on zeta(s+it) for fixed s

Answer by engelbrekt for Lower bounds on zeta(s+it) for fixed s