I was randomly browsing, when I found this puff piece claiming a proof of the Lindelöf hypothesis by Fokas. Note that the Wikipedia article says that he claimed, then withdrew his claim in 2017, but the USC piece is dated June 25 2018. So, what is the truth?

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    There is a revised Arxiv print of his dated June 19 2018. The truth is that Wikipedia does not update as quickly, and I imagine some are waiting for expert review. Gerhard "Puffs Wait For No One" Paseman, 2018.06.28. – Gerhard Paseman Jun 29 at 2:13
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    As I read it, the June 19 version does not appear to claim that he has proved the Lindelöf hypothesis. "Hence, since the above identity is valid for all $\epsilon$, this asymptotic identity suggests the validity of Lindelöf's hypothesis". – Robert Israel Jun 29 at 6:22
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    @RobertIsrael He certainly does claim it everywhere else, it seems :) – Igor Rivin Jun 29 at 13:23
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    I happened to attend this colloquium talk at UMass Amherst by Fokas in March 2018: math.umass.edu/calendar/distinguished-lecture/17401. During the talk he definitely claimed that the proof of the Lindelöf hypothesis was forthcoming (some parts joint with coauthors); apparent he had already achieved a "formal derivation" of LH in some sense but still needed more hard analytic work to rigorously verify this derivation. (I know nothing of this area so my memory/paraphrasing could be way off.) – Sam Hopkins Jun 29 at 22:37
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    @SamHopkins: That colloquium talk was on March 29, that is, between versions 3 and 4 of his arXiv preprint (arxiv.org/abs/1708.06607). In version 3 he says "using the fact that [...] the lhs of (1.16) satisfies the Lindelöf hypothesis, it is possible to show that the Riemann zeta function satisfies the same hypothesis. [...] rigorous details are provided in [FKL]." In version 4 he says that "(1.6) suggests the validity of Lindelöf's hypothesis", and he no longer claims that the proof of the LH is forthcoming. – GH from MO Jun 29 at 22:49

(Not an answer of any sort, just too long for a comment.) The main result seems to be an integral equation (1.3) of the form $$\int_{-\infty}^\infty K(t,\tau) |\zeta(\tfrac{1}{2}+it\tau)|^2\,d\tau={\mathcal G}(t)$$ with some explicit functions $K$ and ${\mathcal G}$. This equation (if true) is presumably new, and may be interesting.

However, in my view, how interesting it is would depend quite a bit on whether $|\zeta(\tfrac{1}{2}+it\tau)|^2$ is the only solution of it. This sort of integral operators may have kernels, and if it is the case then it would be rather difficult to squeeze the Lindelöf hypothesis out of it. (If I were the author then this is where I would look.)

P.S. For those who have read ``puff piece'': As pointed out by Robert Israel, indeed, in the (this far, latest) version 4 a proof of the Lindelöf hypothesis is not claimed.

P.P.S Take the above with a pinch of salt; the last time I was involved with this subject was decades ago.

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    Some are not too hopeful with respect to this approach... – Andrey Rukhin Jun 30 at 14:36

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