TITLE: Bounds on horizontal minima of the Riemann zeta function
QUESTION [8 upvotes]: It is known that $\zeta(s)$ has an infinity of zeros in the strip $0<\sigma<1$ and that those zeros become closer together as $t\rightarrow\infty$. More precisely, Littlewood showed that there is a zero $\rho=\beta+i\gamma$ with $|\gamma-t|\in O(1/\log\log\log t)$. 
A positive proportion of these zeros are known to have $\beta=1/2$, but $\zeta(1/2+it)$ is unbounded. In fact, $\zeta(\sigma+it)$ is bounded as $t\rightarrow\infty$ only if $\sigma>1+\delta$, $\delta>0$ fixed.
On the other hand, there seem to be good reasons to expect that
$$f(t)=\min_{0\leq\sigma\leq 1} |\zeta(s)|$$
approaches zero as $t\rightarrow\infty$. One reason is that zeros of $\zeta(s)$ are rather differently distributed than those of $\zeta(s)-z$ for any other value of $z$. 
I would like to argue that $f(t)\rightarrow 0$ here, but my approach is incomplete. In fact, I would like to conjecture that, for every $\epsilon>0$, we have $$f(t)\in O (\log^{\Theta(t) -1+\epsilon}t),$$ 
where $\Theta(t)=\max\{\beta:\gamma<t\}  $. 
My questions are as follows: is anything at all non-trivial known about upper bounds on $f(t)$ and/or who has studied it? Does anyone have any computational data supporting this conjecture? Thanks!

REPLY [2 votes]: The following paper deals with the behavior of the modulus of the Zeta function in the critical strip along horizontal lines:

MR1986257 (2004c:11152)  Saidak, Filip ;  Zvengrowski, Peter . On the modulus of the Riemann zeta function in the critical strip.
Math. Slovaca  53  (2003),  no. 2, 145--172.

See also the slightly related paper

MR2857985 (Reviewed)  Srinivasan, Gopala Krishna ;  Zvengrowski, P.  On the horizontal monotonicity of |Γ(s)|.
Canad. Math. Bull.  54  (2011),  no. 3, 538--543.

See also arXiv:1205.2773