Comment by impendia | Hacker News Reader

Comment by impendia | Hacker News Reader

Speaking as an analytic number theorist, the branch of math of which the Lindelof Hypothesis is part:

This is a huge deal, if true. But USC's PR machine seems to have jumped the gun.

The paper in question, found here

https://arxiv.org/pdf/1708.06607.pdf

has so far only been posted to the arXiv (and only eight days ago). It has presumably not been subjected to any sort of peer review yet. No third party other than USC has announced the results. There's no chatter among my mathematician friends, or on the blogosphere.

Fokas's results could be correct. If the community comes to a consensus that they are, this would be a tremendous advance, and the analytic number theory community as a whole will be trumpeting them.

But, for the time being, I stipulate that some small technical error is probably lurking in the details, which would take hours to find, and which will tank the proof.

I hope that I am proven wrong. Until then I propose the headline: "Mathematician-M.D. claims to have solved one of the greatest open problems".

I just skimmed through it so maybe I'm missing something, but it also appears to be incomplete. It even says "the completion of the rigorous derivation of the above results will be presented in a companion paper". He brings up equation 1.6 but I don't see a rigorous handling of its asymptotics, nor any precise statement of what the growth rate of zeta on or near the critical line is expected to be. The conclusions section also says about as much. I hope it works out and we will see the conclusion, but I'd really like to see some precise growth rates so we can at least compare it to known bounds and example.

There are earlier revisions of that arxiv submission from 2017. I think if it had introduced an idea that can prove a problem this hard, it would have already been creating buzz. Otherwise I don't know what to make of its submission history.

[nevermind]

The work of Athanassios Fokas is hardly unnoticed or scoffed at, in general.

Sure. We've put that title above.

In my university,my professor told us about the existance of this proof in December,and that he gave a presentation that time.So i guess it has been reviewed thoroughly.

[EDIT: edited this comment substantially.]

Speaking as an analytic number theorist, the branch of math of which the Lindelof Hypothesis is part, could you tell us what you personally think of the paper?

(Let's try to avoid just gratuitous negativity: https://blog.ycombinator.com/new-hacker-news-guideline/ )

I read the paper after you linked it and it seems serious. Fokas is the guy who invented the Fokas Method - https://en.wikipedia.org/wiki/Fokas_method

Structurally this is what my impression is of the paper (I don't understand the math):

The result seemed very well-presented, includes background and a 2-page inroduction, summarizes the derivation of the main result from page 5-12, derives its main theorems and lemmas used, and then from p. 50-52 summarizes it all again. Finally three appendices in 4 pages provide some numerical verification (a sanity check) and an acknowledgment section says "This project would not have been completed without the crucial contribution of Kostis Kalimeris. Kostis has studied extensively the classical techniques for the estimation of single and multiple exponential sums; these techniques are used extensively in our joint paper with Kostis [KF] and some of the results of this paper are used in section 6. Furthermore, Kostis has checked the entire manuscript and has made important contributions to the completion of some of the results presented here." There are another 7 paragraphs of acknowledgments going back more than 3 years, and finally 3 pages of references, including to private correspondence and preprints.

The affiliations on the paper are Department of Applied Mathematics and Theoretical Physics, University of Cambridge, and Viterbi School of Engineering, University of Southern California.

Additionally, this researcher has a proven track record, his Wikipedia article says:

>He has made seminal contributions in a remarkably broad range of areas which include: symmetries, integrable nonlinear PDEs, Painleve' equations and random matrices, models for leukemia and protein folding, electro-magneto-enchephalography, nuclear imaging, and relativistic gravity. Also, he has introduced a completely new method for solving boundary value problems known as the Fokas method, which has been acclaimed as the most important development in the analytical treatment of PDEs since the introduction of the Fourier transform

He is the winner of the Naylor prize (past winners include Roger Penrose, 1991 and Stephen Hawking, 1999, among others) and holds 7 honorary doctorates (right side of his Wikipedia page.)

No, it is the proper response to unverified claims of scientific breakthrough.

> This is a huge deal, if true.

What's the potential benefit to humanity that could come about if this is true, without any other dependencies, over the next 30 years?

That's just not how pure mathematics works.

For instance, imagine asking that same question about Euler's totient function 50 years before RSA was developed.

I am sure that there are countless other examples where a seemingly useless theorem aided in a practical problem decades or centuries later.

Now I and other non-mathematicians can look into Euler's totient function and RSA, and learn more. Thanks for answering.

So is there really never a proof in pure mathematics with more obvious or immediate near-term applications?

Are there any historical examples of that? A major breakthrough in mathematics that once understood, it was immediately obvious that it was going to make X work better, and then it did?

Mathematics with immediate payoff is usually called physics or computer science. For example the simplex method for linear programming made a lot of things more efficient.

The maths that has an immediate near-term application usually falls, almost by definition, under applied mathematics (or possibly a related field, such as physics, engineering, computer science etc.).

Pure mathematics essentially concerns itself with the mathematics of mathematics. It's basically the process of trying to either prove certain "empirically discovered" mathematical facts, or understand deeply what those facts mean in a more general way. Understanding this type of mathematics in this deep way allows further mathematics to bud off of that specific

A deeper understanding of pure mathematics? That's like asking what benefits Michelangelo's works had for humanity.

I mean... not quite...

Unless you're talking about improvements in artistic techniques, the art itself doesn't really lead to something more practical, whereas pure mathematics actually does often lead to applications in other fields.

Modern mathematics is so incredibly abstract that I'd be very surprised if more than a minuscule fraction of all published theorems ever had practical applications. If you count that you should also count improvements in for example chemistry (pigments) or the theory of human perception that were initiated by art.

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