Let $\zeta$ be the Riemann zeta function.


My question is: For fixed $\sigma<1/2$, how large can $|\zeta(\sigma+it)|$ be for $t\in \mathbb{R}$, even assuming zeta conjectures like the RH or the LH ? 


My searches in relevant texts like Titschmarsh reveal that people seem to be focused on the case $\sigma= 1/2$, for which Lindelof conjectured that $|\zeta(1/2+it)|\ll t^{\epsilon}$ for any $\epsilon>0$.



How large can $|\zeta(\sigma + it)|$ be for $\sigma<1/2$?
I answer here for what is known for general $\sigma$, without assuming anything (in particular not LH). Let $s = \sigma + it$ where $s$ and $t$ are real numbers.


**If** $\mathbf{\sigma > 1}$, then the absolute convergence of the Dirichlet series yields
$$\zeta(s) \ll 1 \qquad(\sigma>1)$$
(here and below, the $\ll$-constant may depend on $\sigma$.


**If** $\mathbf{\sigma < 0}$, then we can use the functional equation in order to come back to the known region $\sigma > 1$. Indeed, recall that $\zeta(s) = \gamma(s) \zeta(1-s)$ for a certain explicit completing factor $\gamma(s)$. It can be written in terms of $\Gamma$ functions and the Stirling formula easily leads to
$$\gamma(s) \ll\_\varepsilon |t|^{\frac{1}{2} - \sigma + \varepsilon}$$
Since $\zeta(1-s)$ is bounded for $\sigma < 0$ by the previous case, we deduce
$$\zeta(s) \ll |t|^{\frac{1}{2} - \sigma} \qquad(\sigma<0)$$


**If** $\mathbf{0 \leqslant \sigma \leqslant 1}$, the Phragmén-Lindelöf principle allows to interpolate the bounds above in the critical strip, so that
$$\zeta(s) \ll |t|^{\frac{1}{2}(1 - \sigma)}\qquad(0<\sigma<1)$$


Good references for such matters are for instance Montgomery-Vaughan, *Multiplicative Number Theory, 1. Classical Theory*, or Tenenbaum, *Introduction to Analytic and Probabilistic Number Theory*.


As for the Lindelöf hypothesis (implied by RH, so this bound holds under RH as well.), it states as you said that for $\sigma = 1/2$ a certain bound is satisfied, namely
$$\zeta(s) \ll\_\varepsilon |t|^\varepsilon.$$


Is you are interested in what happens in the whole critical strip, you can apply the Phragmén-Lindelöf principle as well to sharpen the bound. You get assuming LH (or RH),
\begin{align\*}
\zeta(s) &\ll\_\varepsilon |t|^{\varepsilon}\qquad\left(\frac12\le\sigma<1\right) \\
\zeta(s) &\ll\_\varepsilon |t|^{\frac12-\sigma+\varepsilon}\qquad\left(0<\sigma<\frac12\right)
\end{align\*}


