
“The Riemann Hypothesis” by Michael Atiyah – Preprint - chenzhekl
https://www.cnbeta.com/articles/tech/771037.htm
======
reikonomusha
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](https://www.scottaaronson.com/blog/?p=304)

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

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mlechha
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.

~~~
m00n
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.

~~~
shoo
> 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?

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

~~~
marcinmozejko
And x = 0 is not an open set.

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

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ginnungagap
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.

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eranation
I’m guessing this is related to this?
[https://www.newscientist.com/article/2180406-famed-
mathemati...](https://www.newscientist.com/article/2180406-famed-
mathematician-claims-proof-of-160-year-old-riemann-hypothesis/)

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...](https://mathoverflow.net/questions/311062/sir-michael-atiyahs-
conference-on-the-riemann-hypothesis)

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LifeInSlowLane
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.

~~~
tecleandor
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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pvitz
The claimed proof on one slide in his presentation:
[https://twitter.com/hrnn9107/status/1044143799683944448](https://twitter.com/hrnn9107/status/1044143799683944448)

Edit: And his write-up:
[https://drive.google.com/file/d/17NBICP6OcUSucrXKNWvzLmrQpfU...](https://drive.google.com/file/d/17NBICP6OcUSucrXKNWvzLmrQpfUrEKuY/view)

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JohannFlobuster
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.

~~~
reikonomusha
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.

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sygnon
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.

~~~
techsin101
> 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.

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letitgo12345
Well this is short enough that at the very least, it shouldn't take long at
all to assess the correctness.

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blurrrrr
Here’s the reference [2].

[https://drive.google.com/file/d/1WNbTDKljpUR-4im-
IxqluY1tKer...](https://drive.google.com/file/d/1WNbTDKljpUR-4im-
IxqluY1tKer88-mH/view)

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

[https://www.pscp.tv/w/1zqJVLeqXYDKB](https://www.pscp.tv/w/1zqJVLeqXYDKB)

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ssmmww
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.

~~~
mijoharas
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).

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pg_bot
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](https://www.youtube.com/watch?v=d6c6uIyieoo)

~~~
zwkrt
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...](https://en.m.wikipedia.org/wiki/Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function)
.

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.

~~~
guicho271828
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?

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danbruc
PDF version of the paper.

[https://drive.google.com/file/d/17NBICP6OcUSucrXKNWvzLmrQpfU...](https://drive.google.com/file/d/17NBICP6OcUSucrXKNWvzLmrQpfUrEKuY/view)

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

~~~
aaaaaaaaaab
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