Apparently, assuming the Riemann Hypothesis to be true, we could use Phragmen-Lindelof theorem to deduce the Lindelof Hypothesis. I searched for a while but could not find a proof. Could anyone share a reference for such a proof?
Here is the statement of the theorem.
Phragmen-Lindelof theorem. $s=\sigma +it$. Suppose $f(s)$ is holomorphic in the strip $\sigma_1 \leq \sigma \leq \sigma_2$, except for finitely many singularities, and for all $\epsilon>0$, $$ f(\sigma+it) = O( e^{e^{\epsilon |t|}}), t\to \infty. $$ If $f(\sigma_k+it) = O(|t|^{r_k})$ for $k=1,2$, then $$ f(s)=O(|t|^{r(\sigma)}), $$ where $r(\sigma)$ is the function $\mathbb R \to \mathbb R$ of which the graph is a straight line connecting the two points $(\sigma_k,r_k),k=1,2$.