Originally posted by @sonhouse
https://www.sciencealert.com/top-mathematician-sir-michael-atiyah-solved-a-160-year-old-1-million-maths-problem-riemann-hypothesis
Still to be verified but he already has fields medal and Abel Prize.
I saw the question "
Did the Riemann hypothesis get solved?"
And the answer from Luboš Motl, former Harvard Junior Fellow went:
<quote>
"No, it didn’t. I was excited while expecting the proof because I think that Atiyah’s life-long exceptional mind as well as his specialization within mathematics (especially things like the Atiyah-Singer theorem that could apply to “solutions” of an operator equation corresponding to the zeroes) made him a wonderful candidate to be able to solve the problem. He seems full of energy and he answered my e-mail within 5 minutes on Friday.
But as I discuss in Nice try but I am now 99% confident that Atiyah's proof of RH is wrong, hopeless, his proof doesn’t work at all and the talk – and especially the papers – are full of statements that the experts know to be incorrect.
He has used a special proposed function of the complex variable, the Todd function T(s), which is polynomial in convex regions of the complex plane but non-polynomial elsewhere. Already at this point, I am rather convinced that no such natural function exists. If something is a polynomial in a region, it is a polynomial everywhere. There is no natural canonical extension than the straightforward “analytic continuation”. So I believe that his concept of “weakly analytic” functions is used incorrectly as he sloppily jumps between analytic functions of real and complex variables – which differ in some important aspects.
More importantly, his proof doesn’t seem to use the special character of the zeta function – and its relationship with the sum or Euler formulae or with the distribution of primes – at all. Like 90% of hopeless “proofs” of RH, it seems to use the symmetry properties of zeta(s) in the complex plane only, and claim that out-of-the-critical-axis poles cannot exist. But one may easily modify the zeta function by removing some “well-behaved” zeroes on the critical axis while adding quadruplets of “wrong zeroes”. Atiyah’s proof seems to apply to this “modified” zeta function as well – and claim that it can’t have off-the-critical-axis poles. But by construction, it has such poles.
There are lots of other thought-provoking but deeply misguided claims in the papers. He claims to derive the fine-structure constant in the physics of electromagnetism as if it were similar to “pi” – a renormalized “pi”. Lots of numerologists love to propose some purely mathematical, simple formulae for the fine structure constant which is about 1/137.035999. But the fine-structure constant is a complicated beast that finely depends on properties of all elementary particles, including three generations of quarks and leptons as well as the Higgs boson, and both in quantum field theory and string theory, it’s almost certainly vastly less unique than “pi”. There are many (in the stringy landscape picture, googol-like many) cousins of the Standard Model and each of them has dozens or hundreds of parameters similar to the fine-structure constant. None of them may be assumed to be a towering mathematical constant that is on par with “pi”. And even if this equality between the fine-structure constant and a “simple mathematical constant” held by some miracle, Atiyah hasn’t presented valid evidence backing this miraculous coincidence.
He also makes wrong claims that his proof wouldn’t be allowed in the ZF axiomatic system of set theory. If the complex-analysis proof worked, it would be totally independent of all ZF/GB specific axioms and of the Axiom of Choice. The independence follows from the logical simplicity and down-to-Earth character of the science about the complex functions.
Also, even if the Todd function T(s) existed with the properties that Atiyah promotes and needs, I believe that his proof by contradiction still wouldn’t work. The proof by contradiction starts by assuming the “wrong root” with the smallest imaginary part, and proving that there exists another one with an even smaller one – and that’s a contradiction (if one is the smallest one, there can’t be a smaller one). That’s what he says after the 15-line proof. This is a possible strategy (I actually guessed this strategy correctly in another Quora answer I wrote on Sunday) but there’s no known way to fill the details and reach the contradiction. But his 15-line-long proof doesn’t actually seem to use this strategy at all. Instead, it claims that a function F(s) constructed by composing zeta(s) and T(s) in some way has to be zero, and therefore zeta(s) has to be zero, too. I think that the steps leading towards the claim that F(s)=0 would be algebraically incorrect even if all the assumptions were right.
The papers are thought-provoking and you can see that he has thought about many deep parts of mathematics that are close to mathematical or theoretical physics and he still remembers lots of traces of this vast knowledge but the ways how he combines them these days are completely wrong and irrational, plagued by elementary mistakes in reasoning and algebra and dominated by wishful thinking.
I spent much of Monday with that event and it’s likely that I wouldn’t do it again if Michael Atiyah made a similar big claim. But I still see him as a role model – and if I were offered to be alive at age of 89+ and be this excited about the thinking about mathematics, I would accept the deal."
</quote>
27 Sep '18 04:07
Riemann hypothesis possibly solved:
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