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!



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


