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 In fact, Atiyah’s list of references includes just three works: the aforementioned unpublished paper, his 2018 Abel lecture at the ICM in Rio de Janeiro and Friedrich Hirzebruch’s 1966 Topological Methods in Geometry.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 Science, NBCNews.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: https://mathoverflow.net/questions/263301/what-is-the-current-understanding-regarding-complex-structures-on-the-6-sphere … 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.