However, nosotros were given ii papers that are said to incorporate the proof:
The Fine Structure Constant (17 pages)The 2nd newspaper contains the proof – which would really live on an unproblematic proof accessible to intelligent undergraduates – on xv lines of Page 3.
The Riemann Hypothesis (5 pages)
In the outset paper, Atiyah claims to produce "the Todd function" \(T(s)\) which is weakly analytic together with may live on understood every bit a boundary of analytic functions. Influenza A virus subtype H5N1 representation of the measuring portion is his example. Well, I don't fifty-fifty understand this example. I tin write the measuring portion every bit a boundary of analytic functions of the real variable inward many dissimilar ways (arctan, tanh) but they atomic number 82 to completely dissimilar continuations inward the complex plane! He claims that the transition from the "real analytic" to "complex analytic" is basically unique together with straightforward which is 1 of the things that expect evidently incorrect to me.
In the 2nd paper, he claims to derive a contradiction from the beingness of the smallest (by its imaginary part) root \(s=b\) away from the critical axis but inward the critical strip. In a rectangle going upwardly to this \(b\), he recursively defines \[
F(s) = T(1+\zeta(s+b)) - 1
\] together with using about properties of his Todd portion such as\[
T([1+f(s)][1+g(s)]) = T[1+f(s)+g(s)]
\] he derives \(F(s)=2F(s)\) together with thence \(F(s)=0\), together with because \(F,\zeta\) incorporate a similar transformation "reshaped" past times his \(T\), it would also follow that \(\zeta(s)=0\) everywhere which is wrong. He also claims that this proof would live on an instance of the "search for the outset Siegel zero". Find a "smallest" incorrect root, together with and so exhibit that an fifty-fifty "smaller" incorrect root exists.
Remarkably enough, I was answering the enquiry whether a proof of RH could live on elementary on Quora final nighttime together with I used precisely this strategy every bit a highly hypothetical instance what a simple proof could expect like! In fact, Atiyah claims that the imaginary portion of \(s\) was halved, only similar I wrote! ;-)
In fact, I am fifty-fifty worried that Atiyah was copying from me.
At whatsoever rate, I don't come across how a locally holomorphic portion \(T(s)\) inward the complex plane could obey the properties he needs, together with fifty-fifty if the properties were satisfied, I don't intend that \(F(s)=2F(s)\) follows from them every bit he claims.
Off-topic: Like inward 1945, the City of Pilsen has prepared a state-of-the-art choreography together with the newest music to welcome the U.S. troops who volition liberate us (I said "us", non entirely the "girls", I promise that y'all heard me well) from a totalitarian authorities dreaming well-nigh the European domination. I was really impressed past times the character of this video.
More importantly, patch looking through the papers, I checked whether I couldn't kill the proof past times the same simple declaration every bit the declaration that is plenty to kill 90% of the genuinely hopeless attempts. The genuinely hopeless attempts seem to assume that y'all may only expect at about portion amongst a similar location of the zeros together with poles together with y'all may exhibit that in that location are no nontrivial roots away from the critical axis.
Needless to say, whatsoever such endeavour is incorrect because the properties of the primes, the Euler together with other formulae for the zeta function, or other especial information well-nigh the positions of its zeroes were non used at all. There for certain be some similar functions amongst roots that are away from the critical axis.
And I intend that Atiyah's proof sadly suffers from the same unproblematic problem. He claims that no functions amongst the symmetrically located "wrong" roots be at all – which is clearly wrong. Just direct maintain (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 only removed about ii pairs of zeros \(\rho_1,\rho_2,\bar\rho_1,\bar\rho_2\) from the critical axis together with the numerator added a quadruplet of symmetrically placed (relatively to the existent together with critical axis) zeroes away from the critical axis (I am a perfectionist so I kept the "total reveal of roots" the same to minimize my footprint; amongst about adjustments of \(0.6+0.9i\) above, I could fifty-fifty proceed about moments etc.). I intend that Atiyah's proof, similar hundreds of hopeless proofs, claims that \(\zeta_c(s)\) cannot be at all. But it clearly can, I only defined it. ;-)
(If I call upwardly the 2015 Nigerian "RH breakthrough" well, the guy didn't fifty-fifty direct maintain that.)
Maybe his proof isn't hopeless together with patch constructing the portion \(T(s)\) inward the "fine-structure constant" paper, he is using about especial properties of \(\zeta(s)\) that are non shared past times functions such every bit \(\zeta_c(s)\). But I only don't come across where it could perhaps be.
Instead, what nosotros come across inward the "fine-structure constant" are musings well-nigh the unification of physics together with mathematics that I empathise amongst but how they're presented every bit exact scientific discipline is only champaign silly; addition genuinely crackpot numerology well-nigh the derivation of the fine-structure constant of electromagnetism, \(\alpha\approx 1/137.035999\), from about purely together with canonical mathematical operations that "renormalize" \(\pi\) to a "ž" ("zh") written inward the Cyrillic script, i.e. "Ж". ;-) I really wanted to utilisation a Cyrillic missive of the alphabet inward a paper, too. And this is the most playful together with master copy one.
Sorry, Prof Atiyah, but that made me express joy out loud – together with your comments well-nigh the "well-known Russian letter" inward the verbalise escalated my laughter, together with in all probability those of many who understand or who tin read Russian only fine.
First, \(\pi\) is a mathematical constant so it doesn't change, doesn't run, together with doesn't larn renormalized. On the contrary side, the fine-structure constant of electromagnetism does run together with it is a complicated parameter of the Standard Model that is rather messy together with the constant depends on the "theory at curt distances" (either a quantum land theory or, ideally, a string theory vacuum) addition renormalization flows together with all the renormalization flows depend on the whole electroweak theory (electromagnetism is only a portion of the electroweak theory) every bit good every bit the spectrum of quarks together with leptons, the reveal of their generations, together with all other particles together with interactions of the Standard Model.
The Standard Model is almost certainly non every bit unique together with canonical for its parameters to live on on par amongst \(\pi\). Thank God, Sean Carroll wrote an equivalent declaration a twenty-four hours afterwards me. (I only can't understand how he or whatsoever theoretical physicist could live on uninterested inward the Riemann Hypothesis or "incapable" of next a simple proof of it.) And if Mr Atiyah has believed that \(\alpha\) could live on analogously canonical every bit a \(\pi\) or a renormalized \(\pi\) fifty-fifty a decade ago, together with so I am confident that his contributions to the newspaper amongst Witten well-nigh the \(G_2\) compactifications of M-theory together with the topology modify were at most purely technical, similar those of a graduate student, but he couldn't perhaps write anything right well-nigh the "big picture" of that newspaper because he's completely confused well-nigh particle physics.
In the Team Stanford picture, the Standard Model is only 1 amidst \(10^{500}\) or so – a googol-like large reveal – compactifications of string/M-theory. Each of them direct maintain its ain parameters similar to the fine-structure constant. So 1 such a constant cannot live on on par amongst \(\pi\). But fifty-fifty if e.g. Team Vafa were right together with the reveal of phenomenologically relevant corners of the stringy configuration infinite were much lower, the selection is withal far from unique together with the fine-structure constant is far from a simple canonical parameter comparable to \(\pi\).
So I direct maintain spent many hours past times next this even out together with expressed my admiration for Prof Atiyah past times those efforts to psyche to him. But I intend that the bubble has burst, about credit has been spent, together with I would in all probability non picket his about other endeavour to accomplish something comparably groundbreaking. I withal admire him for his accumulated contributions to mathematics, mathematical physics, together with the bridges betwixt mathematics together with physics, his dear for simplicity, together with his liberate energy together with mightiness to command much to a greater extent than than only the bladder at the historic menstruum of 89+, but the proof of the Riemann Hypothesis will require to a greater extent than than that.
There is something perhaps interesting well-nigh the foreign Todd portion – which would combine about non bad ideas from Hirzebruch together with von Neumann. But he says that the portion is polynomial inward whole convex regions of the complex plane together with locally holomorphic – but non fully holomorphic. I don't understand how a polynomial portion inward the complex plane could live on nontrivial inward whatsoever feel – together with how its "different" extrapolation than the simple polynomial analytic continuation could live on natural inward whatsoever way. Because I intend that other comments, such every bit those well-nigh the computation of the fine-structure constant, are only champaign silly, it seems rattling probable that in that location won't live on anything right together with clever inward the claims well-nigh \(T(s)\) together with the portion amongst the desired properties in all probability doesn't exist.
The endeavour to derive the contradiction from the "first Siegel zero" past times showing that an fifty-fifty smaller 1 exists is something that I notice potentially promising. It was about 50% of those efforts of mine to examine the RH that were not based on the Hilbert-Pólya program. I intend it cannot live on excluded that a simple proof based on this full general strategy does exist. But I am rather sure that Atiyah's endeavour isn't such a simple proof.