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Riemann Hypothesis proved?

posted by Jason Kottke   Sep 23, 2018

Mathematician Michael Atiyah claims that he’s solved the Riemann hypothesis, one of the great unsolved problems in math, and will deliver a talk about the proof on Monday.

In it, he pays tribute to the work of two great 20th century mathematicians, John von Neumann and Friedrich Hirzebruch, whose developments he claims laid the foundations for his own proposed proof. “It fell into my lap, I had to pick it up,” he says.

The Riemann hypothesis, which is one of the $1 million Millennium Prize problems, deals with prime numbers. Even though it was suggested back in 1859 and “has been checked for the first 10,000,000,000,000 solutions”, no one has yet come up with a proof.

Here’s an educated guess about a part of Atiyah’s proof.

Update: For the hard-core mathematicians in the audience, here is a video of Atiyah’s lecture and a paper containing what looks like a very high-level overview of his solution.

Update: Several people wrote in wanting me to highlight the skepticism that surrounds Atiyah’s Riemann claims.

A giant in his field, Atiyah has made major contributions to geometry, topology, and theoretical physics. He has received both of math’s top awards, the Fields Medal in 1966 and the Abel Prize in 2004. But despite a long and prolific career, the Riemann claim follows on the heels of more recent, failed proofs.

In 2017, Atiyah told The Times of London that he had converted the 255-page Feit-Thompson theorem, an abstract theory dealing with groups of numbers first proved in 1963, into a vastly simplified 12-page proof. He sent his proof to 15 experts in the field and was met with skepticism or silence, and the proof was never printed in a journal. A year earlier, Atiyah claimed to have solved a famous problem in differential geometry in a paper he posted on the preprint repository ArXiv, but peers soon pointed out inaccuracies in his approach and the proof was never formally published.

Science contacted several of Atiyah’s colleagues. They all expressed concern about his desire to come out of retirement to present proofs based on shaky associations and said it was unlikely that his proof of the Riemann hypothesis would be successful. But none wanted to publicly criticize their mentor or colleague for fear of jeopardizing the relationship

Update: Some final thoughts on Atiyah’s failed proof:

Firstly, it’s become clear that the work presented by Atiyah doesn’t constitute a proof of the Riemann Hypothesis, so the Clay Institute can rest easy with their 1 million dollars, and encryption on the internet remains safe. The argument Atiyah put forward rests on his function 𝑇(𝑠) having certain properties, and many have concluded that no function with such properties can exist, including in a comment on our own post from an academic who’s written about the Riemann Hypothesis extensively. Dick Lipton and Ken Regan have written a blog post looking at the detail of how 𝑇 is supposed to behave. According to some sources, Atiyah has stated he’ll be publishing a more detailed paper shortly, but not many are holding their breath.

This is not easy mathematics, and even top-level mathematicians sometimes find their proofs don’t hold together; it’s no surprise that a solution to this problem wouldn’t be found in a few lines of mathematics, and it’s just a shame that this was so built up and sensationalised when it’s becoming obvious that Atiyah didn’t consult with anyone else about his proof before presenting it in a public forum.

And with that, Betteridge’s law of headlines holds up yet again. QED.