On Michael Atiyah and the Riemann Hypothesis

Riemann

Riemann has been back in the news lately, thanks to an announcement that his nearly 160 year old hypothesis might be solved.
Public domain image courtesy of Wikimedia CC.

At the 2018 Heidelberg Laureate Forum (HLF), Sir Michael Atiyah gave a lecture in which he claimed to have found a proof for the Riemann hypothesis. If Atiyah’s proof holds up, then the nearly 160 year problem concerning the distribution of primes will finally have a solution.  It’s on the Clay Mathematics Institute’s list of seven Millennium Prize Problems and just one of those — the Poincaré Conjecture — is listed as solved on the institute’s website. However, the $1 million prize for a proof of the Riemann hypothesis is yet to be awarded, and some folks doubt that the long-open problem is finally solved.

It’s unsurprising that the world is watching Atiyah — and waiting to hear the verdict about the proof he proposed. Atiyah received a Fields Medal in 1966. In 2004, he and Isadore Singer were jointly awarded the Abel Prize for their discovery and proof of the Atiyah-Singer index theorem. Yet many other supposed proofs for the Riemann hypothesis have been proposed, only to fall apart under further scrutiny.

The Clay Mathematics Institute’s “Official Problem Description” for the Riemann hypothesis is 11 pages long (with nearly two full pages of references), while Atiyah’s current write-up of the proof is five pages long. “Atiyah attributes much of the theoretical work that underpins the proof to a paper of his own that has been submitted to the Proceedings of the Royal Society A,” Frankie Schembri wrote for Science. “That paper has yet to be published,” she added. In fact, Atiyah’s list of references includes just three works: the aforementioned unpublished paper, his 2018 Abel lecture at the ICM in Rio de Janeiro and Friedrich Hirzebruch’s 1966 Topological Methods in Geometry.

News of Atiyah’s claim has reached far and wide since his announcement. Articles about the development appeared in Science News, Gizmodo, Popular ScienceNBCNews.com, the Irish Times and more. Bloggers also covered the announcement, including Katie Steckles and Christian Lawson-Perfect for the Aperiodical, John D. Cook for his consulting blog, as well as Dick Lipton and Ken Regan for Gödel’s Lost Letter and P=NP.

John Baez also wrote a lengthy thread about Atiyah’s claimed proof on Twitter, starting on September 23 (the day before Atiyah’s lecture at the HLF). He wrote “I bet that Atiyah’s claimed proof…will not convince experts. In 2017 he claimed to have a 12-page proof of the Feit-Thompson theorem, which usually takes 255 pages: https://www.maths.ed.ac.uk/~v1ranick/atiyahtimes2017.pdf. He showed it to experts, and… silence.” What’s more, “In 2016 Atiyah put a paper on the arXiv claiming to have solved a famous problem in differential geometry. The argument was full of big holes: So, I’m not holding my breath this time. But of course I’d be happy to be wrong,” Baez added.

Steven Strogatz tweeted “Uh oh. I have a bad feeling about this. Famed mathematician Michael Atiyah claims proof of Riemann hypothesis,” along with a link to the September 21 New Scientist article about Atiyah’s claimed proof. When asked by Twitter user Jonathan Horrocks “Why bad? I take anything Atiyah says very seriously,” Strogatz responded “I take him seriously too. It’s the same bad feeling I have when a diver attempts an extremely difficult dive: afraid, yet hoping for success.”

The skepticism surrounding the claimed proof doesn’t appear to phase Atiyah. “Nobody believes any proof of the Riemann hypothesis, let alone a proof by someone who’s 90,” he said to Gilead Amit​, author of the New Scientist article. “People say ‘we know mathematicians do all their best work before they’re 40,'” Atiyah said, adding “I’m trying to show them that they’re wrong. That I can do something when I’m 90.”

Regardless of whether Atiyah’s proof holds, he has already done something phenomenal: reignited a worldwide conversation about the Riemann hypothesis. That alone is something to celebrate.

What do you think? Share your thoughts in the comments below or reach out to me on Twitter @writesRCrowell! As always, I’m also happy to hear ideas for future blog posts.

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10 Responses to On Michael Atiyah and the Riemann Hypothesis

  1. Yaoming SHI says:

    Hi:

    Nice article!

    (1) I strongly support Prof. Atiyah in making the public announcement of his proof of RH. It is obvious that his result is probably the best among the authors who are 89 or over.

    (2) Even if it is not considered as a complete proof, nobody can deny the possibility that some future generation mathematicians may find some hints from his presentation and preprints.

    (3) I really think that Prof. Atiyah’s goal is to help to advance the understanding of mathematics from human being as a whole. In this sense, he does not really care that he lost some fame (field medal winner also made mathematical mistakes) in this event.

    (4) I myself have completed a proof of Riemann Hypothesis as well. I submitted to Annals of Mathematics for publication in June 2017 but have not heard from them yet. I also uploaded it to arXiv at https://arxiv.org/abs/1706.08868.

    (5) Because I am not a professional mathematician, not to mention a number theorist, I have hard time convince several professional mathematicians to give me the benefit of doubt and review the first 5 pages of section 2 of my preprint.

    (6) But I do have proper training in physics and chemistry. Here is my old web page at UC Berkeley
    http://www.cchem.berkeley.edu/jehgrp/yms

    (7) I have been wondering the following question: Why do we not hear any new story like that of Ramanujan and Hardy any more?

    (8) I consider myself as epsilon times Ramanujan where epsilon =10^{-10}. I am looking for my (Prof.) Hardy.

    (9) Could you please help me by giving me the benefit of doubt and reviewing the first 5 pages of section 2 of my preprint? If it make sense to you, then you may continue to read the next 5 pages…

    (10) And I wish that eventually you can help me by convincing several other professional mathematicians to review the first 5 pages of section 2 of my preprint?

    Best regards-
    Yaoming SHI

    P.S. here is my old web page at UC Berkeley
    http://www.cchem.berkeley.edu/jehgrp/yms

  2. Shecky R says:

    Given what a lot of folks in-the-know are saying doesn’t seem likely Atiyah’s proof will withstand scrutiny, but interesting that his original approach crosses over to physics. In recent years new connections of prime numbers to physics have been noted, and with prime numbers as the building blocks of math, and math as the foundation of physics, such approaches seem potentially fruitful.

  3. Jimmy Bof says:

    The CV of a proposer of a mathematical proof is an irrelevant social factor with respect to the validity of the proof. In this case, the proposed proof is not valid.

  4. Stephen Crowley says:

    Jimmy, you are right, his ‘proof’ makes even less sense than mine. I did not claim to prove it though, merely came up with another conjecture which if true, would imply the RG. My paper is formed in the language of complex dynamics and I’ve tried to get some feedback, however it is downright asinine the arrogance and hostility that I’ve been treated with by editors at reputed journalists. I was simply wanting to generate new ideas on the topic and was told that because i dared to repeat know definitions for clarity that my paper was somehow invalid. I would really appreciate any feedback someone could give me on my ideas. My paper can be found at http://vixra.org/pdf/1702.0273v5.pdf

    Thanks,
    Stephen crowley

  5. Stephen Crowley says:

    I would appreciate any comments regarding

    http://vixra.org/pdf/1702.0273v5.pdf

    A Sequence of Cauchy Sequences Which Is Conjectured to Converge to the Imaginary Parts of the Zeros of the Riemann Zeta Function
    Authors: Stephen Crowley

    A sequence of Cauchy sequences which conjecturally converge to the Riemann zeros is constructed and related to the LeClair-França criteria for the Riemann hypothesis.

  6. Richard Eicholtz says:

    There is a function that applies to all prime numbers. Every prime number reduced to a single integer by addition will become one of 6 integers 1,2,4,5,7,or 8. you will never obtain 3,6,or 9.these 2 sets total 45,where 18/45=2/5 and 27/45=3/5. It turns out that 3/5 represents infinite time results,and 2/5 represents finite time results. Both are needed to solve the Riemann Hypothesis. The real Pi gives us a infinite continued fraction that makes application of these results, both infinite and finite. I sent some of this information out to some in mathematics,and never heard from them. Anyway let me know what you think. Richard Eicholtz

  7. Willie Johnson Jr. says:

    The solution to TRH is really rather simple. Solution: in the zeta function s=0. This gives precisely negative one half. This is the critical line. The negative one half critical line appears on the positive side of the x axis because the critical line is equal to i=square root of negative one which is equal to one half of negative one. In short the critical line is the diameter of the polholde. It’s that simple. It’s all proven in the books Gyroelectrodynamics and Quaternion Eulerian Calculus by Willie Johnson Jr.

  8. Stephen Crowley says:

    Willie, that sounds interesting , I was where can one find a pdf of the above if you dont have $35 to spend? Incidentally I was working on an invention involving a pair of scissor-paired control-moment-gyros and was revisiting the physics lessons on the conservation of angular momentum as it pertains to the Hamilton-Jacobi equation that I need to solve to generate the sequence of pulses to send to the motors..

  9. Allan Dobbins says:

    It sure brings ‘me out of the woodwork!

  10. Antônio Carlos mota says:

    I think What Atiyah starting of fine structure constant with the value is the shape and pattern of several things in the universe,as the shapes and pattern of prime Numbers are Vera interesting

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