I use SDMB as my primary news source but see no mention of the fact that Sir Michael Atiyah has claimed to have proved RH. Google News doesn’t mention it in its headlines, so my non-mathematical friend had to inform me! (I even lost a bottle of beer to him saying “Nah, SDMB would have told me if there were any serious claim to a proof of RH; I’ll bet a beer you’re misled!”)
For those who don’t follow mathematics RH (Riemann Hypothesis) is the most famous of the math problems for which $1,000,000 prizes have been offered.
I wasn’t sure whether to post this in GD or GQ. If it were in IMHO I’d have to mention my own uninformed opinion, which is that Atiyah’s paper reads just like rambling amateurish crackpottery. (The paper is about deriving physics’ FSC as a fundamental math constant! It doesn’t mention RH which supposedly is a simple corollary of the theory he developed for FSC.)
As I say: It seems like crackpottery and Google suggests that other mathematicians think this is an old old man who has finally published one paper too many.
On the other hand, Atiyah has been called the world’s greatest living mathematician.
Sure sounds the same way to me. But I’m only here to mention that Sir Arthur Eddington once claimed that the Fine Structure Constant was exactly 1/136. Somehow this was related to there being precisely 136×2[sup]256[/sup] electrons in the universe. Later, it was found that the FSC was closer to 1/137. Eddington changed his proof to show that the FSC was exactly 1/137. For that, he earned the name “Sir Arthur Adding-One”.
In the linked paper, Sir Michael writes:
The number Ж is a mathematical candidate for 1/α. It satisfies the criteria of both Feynman and Good, restores the reputation of Eddington and …
…
Ironically, Eddington was later laughed out of court, when 137 was found to need a long string of corrections. In fact these corrections just arise from the iterative process that defines Ж, so Eddington’s two mistakes cancel each other out.
IIUC the fact that he has found a simple math expression that evaluates to empirical FSC (α) with 9 digits accuracy is itself rather amazing.
He then computes Ч via the limit of a sum of an integral. I tried punching it into Wolfram Alpha, but it timed out :).
Well. Surely we can compute it to a few hundred places, and then see if it matches reality once a few more digits are measured. I’m sure it’ll only take a few million Feynman diagrams this time…
Equations like " Ч/γ = Ж/π " which introduces, redundantly, two constants — in Cyrillic alphabet no less — make me wonder if this is all some elaborate practical joke. :smack:
The FSC = 137.03599917 is known empirically to ten digits: that final ‘7’ is ‘±4’. Atiyah claims to have computed Ж to 9 digits with an exact match to empirical FSC — too close to be coincidence. Has anyone checked Atiyah’s arithmetic?
[off-topic] Koide’s formula is another unexplained numeric coincidence involving physical constants. A relationship among the masses of electron, muon, and tau gives 2/3 = 0.66666, exact within the uncertainty in those mass measurements. But that 5-digit match might be by chance. Atiyah claims a 9-digit match with FSC.
There is also the lovely result that from the von Neumann hyperfinite factor A of type II and the Hirzebruch Formalism, we can renormalize the usual Euler formula e[sup]2πi[/sup] to e[sup]2Жw[/sup]. It merely requires introducing the new constant w = πi/Ж.
This is the first I have heard of the RH having been proved. Sir Michael Atiyah is a great mathematician, arguably among the top five of the 20th century, but his specialty was differential geometry, not number theory and I suspect that the top number theorists of my department would be talking of little else if it were true. On this basis, I have to doubt it.
I have heard that the abc has been claimed to have been proved (a technical conjecture in number theory that would readily prove the Fermat conjecture and much more) and the proof has been claimed to be wrong and the parties have not come to a resolution. You might thing this sort of thing doesn’t happen in math, but it does. The first purported “proof” of the 4 color theorem stood for 10 years before the error was found.
But I will ask around and see what people are saying.
He’s not some crank with a theory, he is a well respected mathematician with quite a bit of credit to his reputation.
That said, it is a complicated proof, and it may take quite a while to find an error in it, and quite a bit longer to fix the error.
Fermat’s last theorem went through the same thing, with people thinking that they had solved it, only to find a small flaw in their hundreds of pages. that did eventually get proved, but it took a bit longer.
Also, my understanding is that it is a proof by contradiction, which some mathematicians don’t consider to be a “real” proof. It means that if you assume the Riemann hypothesis to not be true, then you get logical contradictions that he shows with his work.
Why in the world would a proof by contradiction not be a real proof. Cantor’s diagonal proof is a proof by contradiction, and I think mathematicians consider it to be a real proof.
Could someone please summarize what the fine-structure constant has to do with the Riemann hypothesis? Especially if it’s not in the paper. Does it have to do with Hilbert’s old idea of realizing the zeros of the ζ function as eigenvalues of some self-adjoint operator?
He is a well respected mathematician but not a physicist and a lot of that manuscript is making claims about solving random physics, music etc…without looking at a single property of the Riemann zeta function .
The lack of a real proof and the kitchen sink nature of that paper should be a red flag but here are a few hit on the Physics Crackpot Index too.
The overwhelming consensus of qualified mathematicians is that it proves nothing.
Don’t take the fact that I linked to something called the crackpot index to tarnish his earlier work. Unfortunately it appears that he is suffering from what we all will suffer from one day, mortality.
Perhaps we are just missing the brilliance but the odds are not pretty remote.
I don’t think it’s that popular, but adherents of intuitionist logic reject the law of the excluded middle, and so don’t get much mileage out of proof by contradiction.
Nothing that anyone can tell. Unfortunately it seems to be embarrassing chapter of an aging mathematician. The lack of public denouncement by others in the field seems to be due to respect for the man.
Apparently he started mentioning his work on a grand unifying theory shortly after his beloved wife passed.
If you watch the presentation this may be a bit more clear.
The Riemann Hypothesis is probably well known enough that when it is solved… well and truly solved… it will be broadcast far and wide enough so that millions of people who have never heard of the RH will quickly know it’s been solved.
The “millions” who find out won’t be random, they’ll be partially self-selected by being newspaper subscribers, journalists, bloggers, googlers, members of forums
cryptographers, students, etc who will be exposed to the news via their own sources.
But, yeah, like Fermat, the millions who learn will be outweighed by the billions who won’t.
That’s probably Serre, but Atiyah is ridiculously brilliant and accomplished. (I’m admittedly biased, since one of the areas I’m in, topological K-theory, is one in which Atiyah is a major figure. He’s also a great author.)
That having been said, this paper honestly does look like something a crackpot would come up with. It reads like a half-remembered summary of another paper the author read somewhere. Math papers (and these look like notes rather than a real paper) methodically go through points in order and detail; this looks like notes a student would take at someone else’s lecture. I can’t follow what’s going on in it, primarily because there is nothing going on. It’s just gibberish.
It also pings nearly every crackpot index: lots of discussion of philosophy and grandoise claism without anything to back them up; lots of discussion of math, but few (if any) actual equations and results; analogies to historical results (Archimedes’ formula for the area of a circle) that have nothing to do with the topic at hand; and lots of promises of awe-inspiring results with with nothing to back them up. Weird, considering that Atiyah is a legimate genius; it would be hard to come up with someone with a better reputation among living mathematicians.
I should clarify above that I was referring to the notes in the OP about the fine structure constant. I did run across a similar 5-page blurb from Atiyah about his purported proof of the RH, and it’s bizarre. At one part, for example, he constructs a complex function F, notes that F(s) = 2 F(s), and then concludes that “Since C is not of characteristic 2, it folows that F(s) is identically 0.” That’s insanely obvious and not something you need to explicitly state, and it’s true even in characteristic 2 (where the equation above literally becomes F(s) = 0); just subtract F from both sides. He goes on to conclude that a certain polynomial T “is not the zero polynomial and so it is invertible in the field of meromorphic functions.” I mean…sure, anyone reading this paper knows what ‘meromorphic’ means, and that’s patently obvious. Weird. It’s also odd in that crackpots tend to have the same combination of obsessive, pedantic detail about completely obvious things and hand-waving carelessness about nontrivial, important things. Atiyah’s not a crackpot, but these notes are just as bad as any internet crank’s.
He’s an amazing guy (and, indeed, everything I’ve seen so far on various social media have addressed the matter much more kindly than they would if it were some random nobody), but it unfortunately looks like he’s lost it.
saying “Nah, SDMB would have told me if there were any serious claim to a proof of RH; I’ll bet a beer you’re misled!”)