Latest known result on Lindelöf hypothesis The Phragmén–Lindelöf theorem gives a consequence of the Riemann hypothesis, viz., the Lindelöf hypothesis. As such this is weaker than Riemann hypothesis; but it is still considered that even a proof of this weaker result will be a breakthrough.
Question:

What is the strongest known result yet on the Lindelöf hypothesis?

 A: The best result of one type so far is this from Jean Bourgain:
$$\large{|}\zeta\large(\small{\frac{1}{2}}+\large{it}\large)\large{|}=\mathcal{O}(t^{\frac{13}{84}+\epsilon})$$
$\frac{13}{84}$ = 0.15476..., so not a huge improvement on Hardy and Littlewood's 0.16666... of around a century ago.
A: The Lindelof Hypothesis may have been proved
In work published at the Arxiv (latest version March 2018, previous version November 2017) Professor Athanassios Fokas of Cambridge University states that he has proved it. In the first version of his paper (August 2017), he offers a proof of a "slight variant"; in the second and third versions he says it is possible to get from there to a proof of the Lindelof hypothesis itself. The publication of the required step is stated to be under preparation in a linked paper co-authored by himself and two other researchers.
Given the stature of the claimant, the claim is in a different category from the many that are made by mathematicians not previously known for having published weighty peer-reviewed results who write that they have proved famous hypotheses.
A: The best result currently known along these lines, due to Hermann Weyl, is that
 $$|\zeta(\frac{1}{2}+it)|=\mathcal{O}(\tau^{16+\epsilon})$$ for any $\epsilon > 0$ as $\tau \to \infty$.
You can refer more about this on: http://www.openquestions.com/oq-ma014.htm
