Is there any 'guess' as to how the Riemann zeta function $\zeta(\sigma+it)$ (or its modulus) behaves to leading order as $t\rightarrow\infty$, for fixed $\sigma$ in the critical strip? Obviously this can't be known for sure until at least the Lindelof Hypothesis is solved, but have people come up with *either* a good guess based on numerical data *or* a definite answer with the assumption of the Riemann Hypothesis?

I know the Lindelof Hypothesis (which is a consequence of RH) would give $\zeta(\sigma+it)=O(t^{0.5-\sigma+\epsilon})$ for $\sigma\le0.5$ and $\zeta(\sigma+it)=O(t^{\epsilon})$ for $\sigma\ge0.5$, but I'm looking for something like $|\zeta(\sigma+it)|\sim Kt^{0.5-\sigma+\epsilon}$ or $K(\log t)^{\alpha}t^{0.5-\sigma+\epsilon}$ for some constants $K$ and $\alpha$.

Many thanks in advance for any help with this!

proved; I was just looking for either numerical data or consequence of RH (which implies LH). $\endgroup$ – Harry Macpherson Apr 10 '14 at 17:44