Has the Riemann Hypothesis been proven?
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 nonmathematical friend had to inform me! (I even lost a bottle of beer to him :p 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.) There are two different YouTubes of his recent lecture, which goes into less technical detail than the paper, itself already lacking in detail. Click here to get his voice and the slides or click here to watch him speak but with slides too small to be legible. 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. 
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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 ...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. 
His first relation is:
Ч/γ = Ж/π 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... 
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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? [offtopic] 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 5digit match might be by chance. Atiyah claims a 9digit 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^{2πi} to e^{2Жw}. 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. 
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But, while it is a proof, it is not as satisfying. You can say that it must be true, but you are not really saying "why". There have been proofs that have been updated to have a "better" proof, whether it be simpler or uses fewer assumptions. This could be one where a new proof could be as "exiting" as a straightforward proof that does say "why". 
Could someone please summarize what the finestructure 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 selfadjoint operator?

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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. 
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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. https://www.youtube.com/watch?v=jXugkzFW5qY I wish him the best. 
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.

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The "millions" who find out won't be random, they'll be partially selfselected 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. 
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That having been said, this paper honestly does look like something a crackpot would come up with. It reads like a halfremembered 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 aweinspiring 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 5page 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 handwaving 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. 
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The last few posts have convinced me that it is all likely early stages of dementia. A shame, because he was a great mathematician. 
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I started to itemize issues or required points of clarification with this claim but stopped as it felt like I was stepping on puppies. I'll leave it to others. 
It looks like his purported 12page proof of FeitThompson isn't available anywhere online, but he has a paper about complex structures on S^6 on the arxiv and another in a book called "Foundations of Mathematics and Physics One Century After Hilbert." The former is just nonsense. I can only find a preview of the first two pages of the latter, but they're just elementary, unrelated stuff about Minkowski space, and the entire thing is only four pages long. Oh well.

Sir Michael Atiyah just passed away:
https://www.bbc.com/news/scienceenvironment46850763 More extensive obituary: https://www.nytimes.com/2019/01/11/o...iyahdead.html 
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As an aside, I've long suspected that the finestructure constant can, in fact, be mathematically calculated. It's not like it'd be the first time that a dimensionless constant of fundamental physics has been discovered to be calculable. But at the same time, any claim of a calculation should be treated with skepticism, especially if it doesn't come from a coherent physical model which makes other testable predictions. I'm not convinced by arguments like "this simple formula produces α to ten decimal places, and that can't be coincidence!", because even "simple" formulae generally contain more information than they seem to, and coincidences happen, especially when you have the whole universe of numbers to look for.

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