Michael Atiyah claims proof of 160-year-old Riemann hypothesis

[ewmayer]

Famed mathematician claims proof of 160-year-old Riemann hypothesis | New Scientist

Michael Atiyah - no crank, he, but note the cautionary bit at end of the article:

Atiyah is well aware of this history of failure. “Nobody believes any proof of the Riemann hypothesis, let alone a proof by someone who’s 90,” he says, but he hopes his presentation will convince his critics.

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.

New Scientist contacted a number of mathematicians to comment on the claimed proof, but all of them declined. Atiyah has produced a number of papers in recent years making remarkable claims which have so far failed to convince his peers.

[Uncwilly]

[Dr Sardonicus]

That popcorn is just the thing.

The video of Atiyah's Heidelberg Laureate Forum lecture (49 minutes) is now on line.

RH has become increasingly tantalizing, as its analogues in other contexts (e.g. function fields over finite fields, or the Weil conjectures) have actually been proven.

I didn't have popcorn handy, so skipped through the talk, and didn't get the impression he was talking much about the actual proof. His answer to the first question indicated he had used an indirect argument.

[Mark Rose]

Here is the paper: https://drive.google.com/file/d/17NB...QpfUrEKuY/view

[CRGreathouse]

You'll also need this:
https://drive.google.com/file/d/1WPs...mTQE0Ww4a/view
though I can't make heads or tails of it. Does the limit in (8.11) -- the definition of Ж -- even exist?

[ccorn]

Luboš Motl is rather disappointed:

Nice try but I am now 99% confident that Atiyah's proof of RH is wrong, hopeless:

More importantly, while looking through the papers, I checked whether I couldn't kill the proof by the same simple argument as the argument that is enough to kill 90% of the truly hopeless attempts. The truly hopeless attempts seem to assume that you may just look at some function with a similar location of the zeros and poles and you may show that there are no nontrivial roots away from the critical axis.

Needless to say, any such attempt is wrong because the properties of the primes, the Euler and other formulae for the zeta function, or other special information about the positions of its zeroes were not used at all. There surely exist some similar functions with roots that are away from the critical axis.

And I think that Atiyah's proof sadly suffers from the same elementary problem. He claims that no functions with the symmetrically located "wrong" roots exist at all – which is clearly wrong. Just take [...]

[CRGreathouse]

My assessment is similar: it sure doesn't seem like a proof.

[Dr Sardonicus]

I was discomfited by what I read about his recent claims about a short proof of the Feit-Thompson Theorem (all groups of odd order are solvable), and a supposed proof of a conjecture about the six-sphere. Not least, by the phrase "failed to convince his peers."

Noooooooooo! That's not it. This is supposed to be mathematical proof, not a matter of "convincing" people. If an error is found in the argument, the proof is not valid. It's dead, Jim.

[a1call]

This discussion is well, well above my head.
But generally speaking, I think that Mathematics is not as B & W as it used to be before the advent of the Wikis/Wikipedia. Unfortunately now it is more of a democratic process rather than a matter of logic and proof.
If it is referenced in a Wikipedia article then it is gospel material.

[jasong]

Quick question, then a comment.

Is this that thing about prime numbers resembling crystal patterns?

And the comment:

I haven't examined the math, because I am certain I wouldn't understand it, but how wrong is this dude?

The way I see it, a proof is supposed to be right 100% of the time, that's part of the definition.

That being said, even if he's found a pattern that doesn't always hold up, that's extremely interesting, even if there's something missing from the underlying theory.

Sorry, sometimes people talk to me to get out of mental ruts, so I just thought I'd put in my 2 cents.

[Dr Sardonicus]

At least Sir Michael's paper is short. The man who first proved that the number is transcendental did not fare well in his tourney with another famous (then) unsolved problem. This "proof," available on line, was 65 pages long.

Ferdinand Lindemann (1907): Über das sogenannte letzte Fermatsche Theorem – Sitzungsberichte der mathematisch-physikalischen Klasse der Bayerischen Akademie der Wissenschaften München – 1907: 287 - 352.

It looks to me like Sir Michael might have done better merely suggesting possible new approaches to problems like RH, the Feit-Thompson Theorem, and the six-sphere problem.