Monday, September 24, 2018

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

The live video stream from Atiyah's talk (9:45-10:30) was mostly overloaded but we may already watch a 49-minute-long recording on the Laureates Forum YouTube channel.

However, we were given two papers that are said to contain the proof:
The Fine Structure Constant (17 pages)

The Riemann Hypothesis (5 pages)
The second paper contains the proof – which would really be an elementary proof accessible to intelligent undergraduates – on 15 lines of Page 3.



In the first paper, Atiyah claims to construct "the Todd function" \(T(s)\) which is weakly analytic and may be understood as a limit of analytic functions. A representation of the step function is his example. Well, I don't even understand this example. I can write the step function as a limit of analytic functions of the real variable in many different ways (arctan, tanh) but they lead to completely different continuations in the complex plane! He claims that the transition from the "real analytic" to "complex analytic" is basically unique and straightforward which is one of the things that look obviously wrong to me.



In the second paper, he claims to derive a contradiction from the existence of the smallest (by its imaginary part) root \(s=b\) away from the critical axis but in the critical strip. In a rectangle going up to this \(b\), he recursively defines \[

F(s) = T(1+\zeta(s+b)) - 1

\] and using some properties of his Todd function such as\[

T([1+f(s)][1+g(s)]) = T[1+f(s)+g(s)]

\] he derives \(F(s)=2F(s)\) and therefore \(F(s)=0\), and because \(F,\zeta\) contain a similar transformation "reshaped" by his \(T\), it would also follow that \(\zeta(s)=0\) everywhere which is wrong. He also claims that this proof would be an example of the "search for the first Siegel zero". Find a "smallest" wrong root, and then show that an even "smaller" wrong root exists.

Remarkably enough, I was answering the question whether a proof of RH could be elementary on Quora last night and I used exactly this strategy as a highly hypothetical example what a simple proof could look like! In fact, Atiyah claims that the imaginary part of \(s\) was halved, just like I wrote! ;-)

In fact, I am even worried that Atiyah was copying from me.

At any rate, I don't see how a locally holomorphic function \(T(s)\) in the complex plane could obey the properties he needs, and even if the properties were satisfied, I don't think that \(F(s)=2F(s)\) follows from them as he claims.



Off-topic: Like in 1945, the City of Pilsen has prepared a state-of-the-art choreography and the newest music to welcome the U.S. troops who will liberate us (I said "us", not only the "girls", I hope that you heard me well) from a totalitarian regime dreaming about the European domination. I was actually impressed by the quality of this video.

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 (the subscript "c" stands for "crippled")\[

\eq{
\zeta_{c}(s) &= \zeta(s)\times R \times \bar R\\
R &= \frac{(s-0.6-9i)(s-0.4-9i)}{(s-\rho_1)(s-\rho_2)}
}

\] The denominators just removed some two pairs of zeros \(\rho_1,\rho_2,\bar\rho_1,\bar\rho_2\) from the critical axis and the numerator added a quadruplet of symmetrically placed (relatively to the real and critical axis) zeroes away from the critical axis (I am a perfectionist so I kept the "total number of roots" the same to minimize my footprint; with some adjustments of \(0.6+0.9i\) above, I could even keep some moments etc.). I think that Atiyah's proof, like hundreds of hopeless proofs, claims that \(\zeta_c(s)\) cannot exist at all. But it clearly can, I just defined it. ;-)

(If I remember the 2015 Nigerian "RH breakthrough" well, the guy didn't even have that.)

Maybe his proof isn't hopeless and while constructing the function \(T(s)\) in the "fine-structure constant" paper, he is using some special properties of \(\zeta(s)\) that are not shared by functions such as \(\zeta_c(s)\). But I just don't see where it could possibly be.

Instead, what we see in the "fine-structure constant" are musings about the unification of physics and mathematics that I sympathize with but how they're presented as exact science is just plain silly; plus truly crackpot numerology about the derivation of the fine-structure constant of electromagnetism, \(\alpha\approx 1/137.035999\), from some purely and canonical mathematical operations that "renormalize" \(\pi\) to a "ž" ("zh") written in the Cyrillic script, i.e. "Ж". ;-) I actually wanted to use a Cyrillic letter in a paper, too. And this is the most playful and original one.

Sorry, Prof Atiyah, but that made me laugh out loud – and your comments about the "well-known Russian letter" in the talk escalated my laughter, and probably those of many who understand or who can read Russian just fine.

First, \(\pi\) is a mathematical constant so it doesn't change, doesn't run, and doesn't get renormalized. On the opposite side, the fine-structure constant of electromagnetism does run and it is a complicated parameter of the Standard Model that is rather messy and the constant depends on the "theory at short distances" (either a quantum field theory or, ideally, a string theory vacuum) plus renormalization flows and all the renormalization flows depend on the whole electroweak theory (electromagnetism is just a part of the electroweak theory) as well as the spectrum of quarks and leptons, the number of their generations, and all other particles and interactions of the Standard Model.

The Standard Model is almost certainly not as unique and canonical for its parameters to be on par with \(\pi\). Thank God, Sean Carroll wrote an equivalent argument a day after me. (I just can't understand how he or any theoretical physicist could be uninterested in the Riemann Hypothesis or "incapable" of following a simple proof of it.) And if Mr Atiyah has believed that \(\alpha\) could be analogously canonical as a \(\pi\) or a renormalized \(\pi\) even a decade ago, then I am confident that his contributions to the paper with Witten about the \(G_2\) compactifications of M-theory and the topology change were at most purely technical, like those of a graduate student, but he couldn't possibly write anything correct about the "big picture" of that paper because he's completely confused about particle physics.

In the Team Stanford picture, the Standard Model is just one among \(10^{500}\) or so – a googol-like large number – compactifications of string/M-theory. Each of them have its own parameters similar to the fine-structure constant. So one such a constant cannot be on par with \(\pi\). But even if e.g. Team Vafa were correct and the number of phenomenologically relevant corners of the stringy configuration space were much lower, the choice is still far from unique and the fine-structure constant is far from a simple canonical parameter comparable to \(\pi\).

So I have spent many hours by following this story and expressed my admiration for Prof Atiyah by those efforts to listen to him. But I think that the bubble has burst, some credit has been spent, and I would probably not watch his another attempt to achieve something comparably groundbreaking. I still admire him for his accumulated contributions to mathematics, mathematical physics, and the bridges between mathematics and physics, his love for simplicity, and his energy and ability to control much more than just the bladder at the age of 89+, but the proof of the Riemann Hypothesis will require more than that.

There is something possibly interesting about the strange Todd function – which would combine some great ideas from Hirzebruch and von Neumann. But he says that the function is polynomial in whole convex regions of the complex plane and locally holomorphic – but not fully holomorphic. I don't understand how a polynomial function in the complex plane could be nontrivial in any sense – and how its "different" extrapolation than the simple polynomial analytic continuation could be natural in any way. Because I think that other comments, such as those about the computation of the fine-structure constant, are just plain silly, it seems very likely that there won't be anything correct and clever in the claims about \(T(s)\) and the function with the desired properties probably doesn't exist.

The attempt to derive the contradiction from the "first Siegel zero" by showing that an even smaller one exists is something that I find potentially promising. It was some 50% of those efforts of mine to prove the RH that were not based on the Hilbert-Pólya program. I think it cannot be excluded that a simple proof based on this general strategy does exist. But I am rather certain that Atiyah's attempt isn't such a simple proof.

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