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Mathematician-M.D. claims to have solved the 110-year-old Lindelöf hypothesis
- princeahmed 8y agohe is a genius renaissance man
- cbames89 8y agoCan someone eli5 the lindelof hypothesis?
- alexbeloi 8y agoNot eli5, but a comparison to the Riemann Hypothesis (RH). RH says the Riemann-zeta function has no zeros along the line (1/2) + iy in the complex plane. The Lindelof hypothesis says that the number of zeros between (1/2) + iy and (1/2) + i(y+1) is much smaller (little-o) than log(y) as y grows. So it can be thought of as a weaker version of RH, but still very very difficult. The fact that Lindelof has been an open problem for over a hundred years (and is an non-trivial weakening of RH) speaks to how difficult RH is as well. Like RH, Lindelof implies things about primes, and also (like RH) has lots of implications about lots of interesting prime-like (irreducible) objects in different spaces.
- dbaupp 8y agoI think you've flipped the condition: the RH says the Riemann zeta function _only_ has zeros along the line 1/2 + iy. (And, indeed, there are known zeros along this line: 1/2 + 14.135... i.) The Lindelöf hypothesis is, apparently, equivalent to: the number of zeros with real part greater than 1/2+epsilon and imaginary part between y and y+1 is o(log(y)), for any epsilon > 0. That is, boxes of height 1 starting just off the critical line contain few zeros; the RH implies they contain zero.
- impendia 8y agoThe Riemann zeta function is the function zeta(s) = 1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + .... For example, zeta(2) = 1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + ... = pi^2/6. As a partially tongue-in-cheek example, zeta(-1) = 1 + 2 + 3 + 4 + 5 + 6 + ... = -1/12. Obviously it doesn't make sense to add all the positive integers (the series doesn't converge), but if you squint and ignore this, and just do the arithmetic a certain way, you get -1/12. The original definition I gave is valid when s is a complex number with real part greater than 1. But the Riemann zeta function can be proved to have analytic continuation: zeta(s) makes sense for any complex number s, other than 1. For example, zeta(-1) really equals -1/12. The zeta function is easy to understand when the real part is greater than 1: the formula I described is enough. Because of the so-called functional equation, it is also easy to understand when the real part is less than 0. But it is in the middle that all of its secrets lie. For example, the notoriously unsolved Riemann Hypothesis stipulates that the "nontrivial" zeroes all have real part 1/2. The Lindelof Hypothesis stipulates that the zeta function grows very slowly along this line (real part = 1/2). It is very closely related to the Riemann Hypothesis. More technical, and of less direct interest to nonspecialists, but in the same family of problems. As an example of how much mathematicians care about this, here are the Google search results for "subconvexity bound": https://www.google.com/search?q=subconvexity+bound https://www.google.com/search?q=subconvexity+bound A "subconvexity bound" is any result which approaches the Lindelof Hypothesis, for either the Riemann zeta function or a more general "L-function". A lot of ink has been spilled on proving results weaker than what Fokas is claiming.
- taneq 8y ago> Obviously it doesn't make sense to add all the positive integers (the series doesn't converge), but if you squint and ignore this, and just do the arithmetic a certain way, you get -1/12. I'm not a pure-maths type person but in my experience, if you get one answer by following simple, well understood maths (like "the sum of two positive integers is a positive integer") and another answer by "squinting and ignoring it", this doesn't mean the simple answer is wrong, it means you did something else wrong (like the hidden divide-by-zero present in your typical "proof that 1 = 2"). Paradoxes point to an error in the formulation of the question.
- denzil_correa 8y ago3B1B has a nice visualization of "Riemann zeta function and analytic continuation" which might help in further understanding of this proof. https://www.youtube.com/watch?v=sD0NjbwqlYw https://www.youtube.com/watch?v=sD0NjbwqlYw
- hsienmaneja 8y agoWhat exactly is the application for cyber security? How does this affect cryptography?
- zitterbewegung 8y agoNothing because it doesn’t show a method about the distribution of prime numbers but it says that given the Riemann hypothesis is true then the leinhoff problem is true .
- robertelder 8y agoI think it's because any information we gain about the Riemann Hypothesis (Lindelöf hypothesis is implied by RH) gives us information about the distribution of prime numbers. Any time you gain information about the distribution of prime numbers you immediately gain information that can be applied to any form of cryptography that makes use of prime numbers. You could use this information either to break existing forms of cryptography faster, or apply it to building newer and stronger cryptography.
- hsienmaneja 8y agoSo, people hire you to break into their places... to make sure no one can break into their places?
- QML 8y ago
- danharaj 8y agoI'm skeptical
- impendia 8y agoSpeaking 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 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".
- danharaj 8y agoThere 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.
- dang 8y agoSure. We've put that title above.
- deleted 8y ago[deleted]
- sometimesijust 8y agoHypothesis: As the complexity of proofs approaches the limits of human ability to understand, saying it is a proof becomes more important than proving it is a proof. Evidence: The Wikipedia page for the Lindelöf hypothesis already unambiguously states that it has been formally proved.
- filmor 8y agoIf you check the history and talk of that page you will see that there is one very persistent user who has repeatedly re-added this section while at least to others tried to remove it.
- hansbo 8y agoIt was a fascinating read. Reminded me of this XKCD: https://xkcd.com/386/ https://xkcd.com/386/
- cbluth 8y agosometimesijust's Law of Proofs
- jvln 8y agoAt the acknowledgement the only big name from the field is Peter Sarnak https://en.wikipedia.org/wiki/Peter_Sarnak https://en.wikipedia.org/wiki/Peter_Sarnak. If Peter Sarnak vouch for the result it might be correct.
- deleted 8y ago[deleted]
- naturalgradient 8y agoJust pointing out that interestingly Cambridge, where Fokas is a professor, has not released anything. He is merely visiting USC so it strikes me as weird that they would claim this PR so quickly. Also Mathematician-MD somehow makes it sound like the MD means he is a lesser mathematician or not a full mathematician. Fokas is a well respected Professor at one of the top applied Maths departments in the world. A better and less biased title would be 'Math Professor' or 'Cambridge math professor' claims..
- princeahmed 8y agohe is a genius polymath . what are you?? just a stupid shit he is a doctor who has a great memory of information and shapes of diseases and he is a great logical person mathematician
- fifnir 8y agoIsn't MD for..like.. medical doctors ? If he's an MD but without a medical degree (which I guess is the case), then what's the difference betweeen MD, PHD or 'professor' ??
- naturalgradient 8y agoSo this goes off on a tangent but I feel it relates to noncentrality [0]. Fokas has a PhD in maths. Being an MD or having gotten an MD 40 years ago is clearly entirely non-central to his career. Calling him Mathematician-MD seems like it is meant to make him seem a lesser mathematician, e.g. by insinuating that this is just something he does part time, and that he can hence be taken less seriously. I don't know what the poster meant by suggesting 'Mathematician-MD', but it reads weirdly to me for that reason. It's highlighting an attribute of a person that is entirely unrelated to his career or this article. Why if not to denigrate him? The title should be changed to neutrally reflect his position. https://www.lesswrong.com/posts/yCWPkLi8wJvewPbEp/the-noncentral-fallacy-the-worst-argument-in-the-world https://www.lesswrong.com/posts/yCWPkLi8wJvewPbEp/the-noncen...
- hypeibole 8y ago