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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asked Jun 29, 2018 at 1:15
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3$\begingroup$ 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. $\endgroup$– Gerhard PasemanCommented Jun 29, 2018 at 2:13
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9$\begingroup$ 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". $\endgroup$– Robert IsraelCommented Jun 29, 2018 at 6:22
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2$\begingroup$ @RobertIsrael He certainly does claim it everywhere else, it seems :) $\endgroup$– Igor RivinCommented Jun 29, 2018 at 13:23
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2$\begingroup$ 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.) $\endgroup$– Sam HopkinsCommented Jun 29, 2018 at 22:37
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9$\begingroup$ @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. $\endgroup$– GH from MOCommented Jun 29, 2018 at 22:49
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3 Answers
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(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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3$\begingroup$ Some are not too hopeful with respect to this approach... $\endgroup$ Commented Jun 30, 2018 at 14:36
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I'm putting this as an answer because I haven't got enough reputation to post a comment.
In a paper published on 25 September 2018, written jointly with A Ashton, A Fokas asserts that a relation derived in that paper "provides the starting point of a novel approach which in a series of companion papers yields a formal proof of the Lindelöf hypothesis" (my emphasis).
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From the announcement of the Séminaire Bourbaki du vendredi on 28/1/2022 :
answered Jan 15, 2022 at 5:10
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5$\begingroup$ This is a survey of new work (presumably mostly about Paul Nelson's preprint from Sep 2021) which has nothing to do with Fokas. It also has nothing to do with Lindelof, in some sense, because it is about extending to more general L-functions things that have been known for $\zeta$ for decades ("resolu par Weyl et Hardy-Littlewood"). So it is clearly not relevant to this (long-dormant) thread. $\endgroup$ Commented Jan 15, 2022 at 8:05
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3$\begingroup$ PS: Chandan, have you perhaps been misled by the Quanta Magazine article on Nelson's work? It is staggeringly misleading, even by Quanta's dubious standards. $\endgroup$ Commented Jan 15, 2022 at 8:06
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$\begingroup$ @DavidLoeffler I wanted to merely point out that Brumley asserts that the Lindelöf Hypothesis is out of reach at present, and therefore indirectly says that it has not been proved by Fokas. $\endgroup$ Commented Jan 16, 2022 at 15:03