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“The Riemann Hypothesis” by Michael Atiyah – Preprint (cnbeta.com)
99 points by chenzhekl 21 days ago | hide | past | web | favorite | 36 comments



The preprint may or may not be from Atiyah (though the writing is consistent with his ramblings about physics and history), but this is an embarrassingly bad “preprint”. It contains almost no substance, and has a myriad of errors of all sorts. It really has many of the usual characteristics [1] of a crank paper, something you can find for a dime a dozen on Vixra.

I know Atiyah is supposed to present on the Riemann Hypothesis at the Heidelberg Laureate Forum on Monday. If the organizers saw this preprint and decided to green-light his lecture, I would consider that disrespectful (to Atiyah & the attendees) and borderline malicious, especially given the context of his other recent mathematical claims, along with his truly bizarre Abel lecture [2].

[1] https://www.scottaaronson.com/blog/?p=304

[2] https://youtube.com/watch?v=fUEvTymjpds


I just went through the preprint and I do not understand your comment. What specifically ticked you off? The preprint is well written, arguments are clear and there's enough background for an expert to work things out.

As Atiyah says in the preprint. The magic is the Todd function and the Mathematical framework that comes with it. It seems Atiyah has developed a new framework (which he calls Arithmetic Physics) and a side product of the framework you get a simple proof of RH. I don't know if the proof is correct. But I don't see any signs of crackpottery in the preprint.

Finally, this is in the style of Atiyah. He is known to be a "theory builder" rather than a "problem solver". True to that, he's claiming a whole new way of looking at number theory. So even if the proof turns out to be false. Mathematicians still get some new ideas.


No, it is not "well written". I'm no expert in analytic number theory, but here are some sanity checks:

His definition of the critical strip (2.4) is wrong.

He works with some family of polynomial functions who agree on the sets K[a] that have open interior (2.1). Of course, two polynomials that agree on infinitely many points are identical. So there really is not much to his "Todd-function". It is just a polynomial.

From his claims 2.3 and 2.4 then follows T(n)=n, for all natural n and hence T(s)=s, as T is a polynomial.

What does "T is compatible with any analytic formula" in (2.4) even mean? Does it mean "for f(X) a everywhere converging power series, then T(f(s))=f(T(s)), for s in C"? This can only hold for T(s)=s, again. So maybe it means something else? He applies it to f(X)=Im(X-1/2), which is not a power series, so what does he mean?

The Hirzebruch reference is a 250pp book. The paragraph on Todd-Polynomials (which are a family of multivariate polynomials, btw. There is no "Todd-polynomial" T in Hirzebruch!) does not contain a formula as claimed in (2.6).

Considering the last two breakthrough claims, that Atiyah made (no complex S^6 sphere and a new proof of Feit-Thompson) vanished in thin air, I remain more than sceptical that this "preprint" can be salvaged.


> He works with some family of polynomial functions who agree on the sets K[a] that have open interior (2.1). Of course, two polynomials that agree on infinitely many points are identical.

Consider f(x, y) := xy and g(x, y) := xy^2

Fixing x=0 note that f and g agree along {(x, y) | x = 0, y in R}. But f is not identical to g. There is no open subset of R^2 such that f and g agree throughout the subset.

Would rewording "two polynomials that agree on infinitely many points are identical" as "two polynomials that agree on any open set" fix this? Or restrict the statement to polynomials in one variable only?


You're right, I was thinking about univariate polynomials. The "preprint" is only concerned with functions of one complex argument, so this should suffice.


C is not R^2. Neither f nor g is a polynomial over C, which is why they aren't a counterexample.


And x = 0 is not an open set.


agreed, x=0 is not an open set for n>1 dimensions, but it is "infinitely many points"


You'll have to go to the previous paper[1], where this Todd function is supposedly defined and...it's not defined there either. I mean, the whole thing is really sad and not something that I feel like talking much. Just give things a read.

[1] https://drive.google.com/open?id=1WPsVhtBQmdgQl25_evlGQ1mmTQ...


Atiyah says, Hirzebruch named and defined the function. Why not look for it in the work of Hirzebruch that Atiyah cites at the end?


Did you do an undergraduate math degree? There are some very real issues with the Todd function, such as no polynomial behaving the way he suggests for pretty elementery reasons. This doesn't even come close to meeting the standards of what a paper should look like, especially one that is claiming such a big result.


It's not a preprint, it's a "teaser" -- the utility of which is suspect to me. It is certainly not the paper he submitted to Royal Society Proceedings. I don't work in this specific subfield but it reads like ramblings. Show the proofs.


I haven't watched the Abel lecture yet. What is it that you find bizarre?


The community should really let this drift away quietly out of respect for a legend, instead of inviting him to conferences. After the S^6 business I'm disappointed by the organisers in Heidelberg.


I’m guessing this is related to this? https://www.newscientist.com/article/2180406-famed-mathemati...

Edit: it seems the internet is saying, yes he is a fields medalist, but hold with the champagne for a minute until this is peer reviewed at least

https://mathoverflow.net/questions/311062/sir-michael-atiyah...



At the end, he basically says it isn't done, nor a formal proof of RH over Q. Secondly, he thinks RH is undecidable in the Godel sense, and I completely agree.

I studied the RH for my Senior Thesis and Godel completeness makes tons of sense here.

In terms of the proof, Proof by contraction has always felt like it yields short proofs. The beauty is the in the assumption and the tools afterwards.

In fact, the more I read the proof, the more beautiful I find the construction to be. Everything falls out. Thats why its so short.

This Todd Function I've never heard of so I need to do some reading.

Seems pretty legit to me but, you need alot of understanding here.

Source: I have a masters in Math and have studied the RH in depth during those studies.


I’m afraid you may have trouble finding the definition of the Todd function; his citation (to himself) doesn’t define it as far as I can tell.

Otherwise, the “proof” here doesn’t really contain a lot. A couple undergrad analysis classes are enough to “understand” (and consequently call out nonsense of) this bit of writing.


If RH is independent, then it's true - since any specific counterexample has a concrete algorithm & proof to locate it.


Well this is short enough that at the very least, it shouldn't take long at all to assess the correctness.



The poor man lost his wife earlier this year and this is not the first time mathematics has seen grand claims coming from someone near the end of their career grappling with extreme grief. I hope we can quietly let this slide without humiliating the legend.


> The poor man lost his wife earlier this year and this is not the first time mathematics has seen grand claims coming from someone near the end of their career grappling with extreme grief. I hope we can quietly let this slide without humiliating the legend.

I think it should be given full review and document crtiscisms of it.


For those that are unaware this is one of the Millenium problems set by the Clay mathematical institute. Its proof carries a million dollar prize along with it. If it holds up, this would be the second out of 7 to have been solved. An explanation of the problem can be found on the Numberphile youtube channel[0]

0: https://www.youtube.com/watch?v=d6c6uIyieoo


If anyone else watched that video feeling they understood the "what" but not the "why" like me, let me try to give an explanation. The Riemann Zeta function is fundamentally the link between the counting numbers and the prime numbers. A striking (if unenlightening) showcase of this link is https://en.m.wikipedia.org/wiki/Proof_of_the_Euler_product_f... .

Counting numbers are the building blocks of addition and primes the building blocks of multiplication. The RH is important because most theorems in number theory pertain either to additive concepts or multiplicative concepts, but rarely both. In some sense there is a fundamental link between addition and multiplication that we still don't understand. One can see the collatz conjecture as a byproduct of this fact. A proof of RH would give insight to what this link is and give us a deeper understanding of why prime numbers seem so regular yet random.


That sounds also related to the ABC hypothesis that a crazy Japanese mathematician published a multi-thousand-page paper on it. Hopefully this clears the thing up in the messed up paper?



Here is an unofficial lifestream of the talk Atiyah gave at the Heidelberg laureate forum:

https://www.pscp.tv/w/1zqJVLeqXYDKB


If this really is Atiyah's claimed proof, it is very sad and embarrassing indeed. I think it is in poor taste to discuss this as if it were a serious attempt at a proof.


Sorry, could you explain some more about this for people with less of a mathematical background? (I have an undergraduate degree in physics, am familiar with the Riemann zeta function, but do not see anything obviously wrong with the paper).


Is there a version of this in English somewhere? I can't read Chinese.


Scroll down.


[dead]


There is some discussion of your paper on math.stackexchange here: https://math.stackexchange.com/a/1900048/24124

I have no idea why you think proving a famous conjecture would endanger your life, but if that is true, HN is probably not able to help. Adding a Hollywood factor to your appeal degrades your work and yourself and is unlikely to generate the kind of attention you want.


What is grammar?


Atiyah obviously has a mental health problem. His purported proof of the Riemann Hypothesis should not be taken seriously. Other recent fiascos include his ludicrous claim of a 12 page proof of the Feit-Thompson Theorem, his asserted proof that there is no complex structure on the 6-sphere, and his talk at the ICM. Folks, these are not minor flubs. There is no resemblance to serious mathematics.

His past achievements are rightly celebrated. Most mathematicians recognize the situation and are respectfully trying to minimize the fuss.


That's what I've heard. They're letting it slide to not make a big deal of it in media.




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