The Phragmen-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?

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.

  • Which result you know how to prove it ? – reuns Aug 25 '17 at 19:14
  • As in my other comment, according to Titchmarsh (pp 97-98 in the edition edited by Heath-Brown) says that the method is due to Weyl, and developed by Hardy-Littlewood. – paul garrett Aug 25 '17 at 20:24
  • @reuns - I'm assuming you mean "Which result? Do you know how to prove it?" The result is the one I mentioned. You will find how to prove it if you click the link I gave to the paper in which it was published. (The mini-abstract on that page gives $\frac{53}{342}$, but if you click on the pdf you will get to the actual paper which gives $\frac{13}{84}$. Then you can read the proof.) – user321773 Aug 25 '17 at 22:52
  • @paulgarrett - Hardy and Littlewood used Weyl's method of estimating exponential sums and applied it to at least one term of the Riemann-Siegel formula. It was they and not Weyl who came up with the $\frac{1}{6}$ figure. – user321773 Aug 25 '17 at 22:59
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    @ruffle, of course, you may be entirely correct... I was not there at the time! But/and I will be interested to look at what Weyl's paper literally did, etc. In the context of subconvexity, I have heard many people (in the last 10-15 years, at least) give Weyl credit for the 1/6, just as they give Burgess credit for the 1/4 in another version. It is always non-trivial to separate mythology from fact, without doing some legwork. Thanks for insisting on this... now I will want to go look at the original sources. – paul garrett Aug 25 '17 at 23:05

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.

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

  • Is $\tau$ the same as $t$? – Matt E Aug 10 '10 at 20:38
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    Weyl didn't work on the Lindelof hypothesis, and there has been a lot of progress since his time. The Wikipedia page is fairly up to date and to do better than that you would need information from someone in the field. "16" is a typo for 1/6, proved by Hardy and Littlewood (not Weyl). – T.. Aug 10 '10 at 21:42
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    This answer is factually wrong, and it would still be wrong if it said $\frac{1}{6}$ and Hardy and Littlewood. I'm not sure why it's still here. In 2010 when it was posted, the correct answer was no larger than Huxley's $\frac{32}{205}$. – user321773 Aug 25 '17 at 16:18
  • Titchmarsh on pages 97-98 says the method elaborated by Hardy and Littlewood was due to Weyl. Indeed, Weyl's 1921 paper with a misleading title about $\zeta(1+it)$ gives ${1\over 6}+\epsilon$. (So, actually, Weyl did effectively work on the Lindelof Hypothesis.) – paul garrett Aug 25 '17 at 20:14
  • I can't prove he didn't, but Wladyslaw Narkiewicz on p.136 of his Rational Number Theory in the 20th Century: From PNT to FLT writes that $\frac{1}{6}$ first occurred in an unpublished paper by Hardy and Littlewood. – user321773 Aug 25 '17 at 23:16

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