Bounds on horizontal minima of the Riemann zeta function 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!
 A: This is not exactly an answer, but too long for a comment.
Perhaps I have not been sufficiently clear in the comments. In the paper:
T. Kotnik, Computational estimation of the order of $\zeta(1/2+i t)$,
Math. of Comp. 73 (2003) 949-956
it is shown how to get values $t$ where  $|\zeta(1/2+it)|$ is large.
Many (almost all) of these values of $t$ are near a Gram point.
The x-ray shows that these Gram points are places where a real line 
(one where zeta is real) hit the critical line. Precisely the real line
goes from $\sigma=+\infty$. On these lines $\zeta(s)$ is real and $>1$.
What the x-ray's show is that the points signaled by Kotnik are those in which
these lines are almost parallel to the real line. So that at these height
$t$ the value $\zeta(\sigma+it)$ is almost real for $0<\sigma<1$. By the election 
of the lines $|\zeta(1/2+it)|>>1$. This is the what  Kotnik is searching, and we know
that $\Re\zeta(\sigma+it) >1$, because these points are almost on the real line. 
Therefore we expect that 
$$\limsup_{t\to\infty} f(t)\ge 1.$$
Of course this contradicts the conjecture above, except if you assume that 
$\Theta(t)\to1$, which contradicts the RH. 
For example I take one of the values given by Kotnik (and he has means to get these
values apparently without limit).
I take $t=21559062801.941668$ where he reported
$Z(t)=-192.996$.  And compute the values 
$$\zeta(it)=410676.057+11579.014 \cdot i$$
$$\zeta(0.1+it)=67487.871+ 1567.236\cdot i$$
$$\zeta(0.2+it)=12335.222+225.398\cdot i$$
$$\zeta(0.3+it)=2584.380+37.231\cdot i$$
$$\zeta(0.4+it)=640.801+7.986\cdot i$$
$$\zeta(0.5+it)=192.982+2.341\cdot i$$
$$\zeta(0.6+it)=71.315+0.841\cdot i$$
$$\zeta(0.7+it)=32.010+0.315\cdot i$$
$$\zeta(0.8+it)=17.005+0.101\cdot i$$
$$\zeta(0.9+it)=10.355+0.010\cdot i$$
$$\zeta(1+it)=7.013-0.027\cdot i$$
As I said the values are 'almost real' and the absolute value 
$|\zeta(1/2+it)|>7$.  But I have taken any of the values of Kotnik.
A: Not a compete answer, but observations on the value of $\sigma$ for which the minimum occurs.  Throughout I'm assuming the Riemann Hypothesis.
In "Zeros of $\zeta^\prime(s)$ and the Riemann Hypothesis", Spira proves that the Riemann Hypothesis holds if and only if $|\zeta(s)|$ is increasing as $\sigma$ decreases from $1/2$ to $-\infty$ for $t\gg 1$ (He seems to prove $t>165$.)  So it suffice to consider the minimum over $1/2\le \sigma\le 1$.
From now on we assume $\zeta(1/2+it)\ne 0$ (else the minimum over $\sigma$ is already $0$ for that $t$.)  Then $|\zeta(\sigma+i t)|>0$, and so $\ln|\zeta(\sigma+i t)|$ is defined.  Since $\ln$ is increasing and preserves inequalities, the same $\sigma$ minimizes both $|\zeta(\sigma+i t)|$ and $\ln|\zeta(\sigma+i t)|$.  There are three possibilities:
I: $\ln|\zeta(\sigma+i t)|$ is increasing on $1/2\le \sigma\le 1$ and the minimum occurs at $\sigma=1/2$.
II: $\ln|\zeta(\sigma+i t)|$ is decreasing on $1/2\le \sigma\le 1$ and the minimum occurs at $\sigma=1$.
III: $\ln|\zeta(\sigma+i t)|$ has a local minimum on $(1/2,1)$.  (Conceivably, the global minimum may still occur at an endpoint.)  This is the case in which
$$
\frac{d}{d\sigma}\ln|\zeta(s)|=0
\Leftrightarrow \frac{\partial}{\partial \sigma} \text{Re}(\log(\zeta(s)))=0
\Leftrightarrow \text{Re}\left(\frac{\zeta^\prime(s)}{\zeta(s)}\right)=0.
$$
This latter happens when either 
IIIa: $\zeta^\prime(s)=0$, or
IIIb: $\arg\left(\frac{\zeta^\prime(s)}{\zeta(s)}\right)=\pm \pi/2.$
The zeros of $\zeta^\prime(s)$ occur discretely (just as do the zeros of $\zeta(s)$).  Meanwhile the level curves for $\arg=\pm \pi/2$ will connect the zeros of $\zeta^\prime/\zeta$ (i.e. the zeros of $\zeta^\prime(s)$) with the poles of $\zeta^\prime/\zeta$ (i.e. the zeros of $\zeta(s)$ on the critical line.)
Titchmarsh shows that $\zeta(s)\ne 0$ for $\sigma>3$, and zeros with $\sigma>1$ do exist.  But Spira shows in "Zeros of $\zeta^\prime(s)$ in the critical strip" that 'most' zeros are close to $\sigma=1/2$ (in the sense that the number of zeros with $\sigma>1/2+\delta$, $t<T$ is only O(T).)  Lots of research has been done on the clustering of the zeros of $\zeta^\prime$ near the critical line.
Edit: For $t\gg1$, possibility I above can not occur. For if it did, then combined with Spira's result above, we would have that $\sigma=1/2$ is a local minimum.  But the formula for the logarithmic derivative
$$
\frac{\zeta^\prime(s)}{\zeta(s)}=\log(2\pi)-1-C/2-\frac{1}{s-1}-\frac{\Gamma^\prime(s/2+1)}{2\Gamma(s/2+1)}+\sum_{\rho}\left(\frac{1}{s-\rho}+\frac{1}{\rho}\right),
$$
along with Stirling's formula, gives that 
$$
\text{Re}\frac{\zeta^\prime}{\zeta}(1/2+it)\sim -\log(t/2)/2,
$$
and in particular is not $0$.
Case II above should correspond to zeros of $\zeta^\prime(s)$ with $\sigma>1$, or consecutive zeros of $\zeta$ with no zero of $\zeta^\prime$ in between (for otherwise the horizontal line $t=\text{const.}$ should cross a level curve $\arg=\pm \pi/2$.)  This latter phenomenon happens at intervals on average of $2\pi/\log(2)\approx 9.064$, due to the different asymptotics of the zeros of $\zeta$ and $\zeta^\prime$.  
Final edit: the graphic below shows $t$ on the vertical axis, and, on the horizontal axis, the Hardy function $Z(t)$ (in purple) and the value of $\sigma$ which minimizes $f(t)$.  Also marked on the vertical axis are the zeros of $Z(t)$ (in black).  The $t$ which correspond to zeros of $\zeta^\prime(s)$ are marked in red when $1/2<\sigma<1$, and in green when $\sigma\ge1$.  [Thanks to Ricky Farr for computation of zeros of $\zeta^\prime$.]  The data illustrates that minima occuring at $\sigma=1$ tend to be associated to consecutive zeros of $\zeta$ with either no intervening zero of $\zeta^\prime$, or a zero of $\zeta^\prime$ with $\sigma\ge 1$.
     (source)
A: 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
