6 ms·

“The Riemann Hypothesis” by Michael Atiyah – Preprint

99 points by chenzhekl 8y ago

eranation 8y ago
I’m guessing this is related to this? https://www.newscientist.com/article/2180406-famed-mathematician-claims-proof-of-160-year-old-riemann-hypothesis/ 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-atiyahs-conference-on-the-riemann-hypothesis https://mathoverflow.net/questions/311062/sir-michael-atiyah...
letitgo12345 8y ago
Well this is short enough that at the very least, it shouldn't take long at all to assess the correctness.
reikonomusha 8y ago
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
- tanderson92 8y ago
  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.
- mlechha 8y ago
  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.
  - vlasev 8y ago
    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_evlGQ1mmTQE0Ww4a https://drive.google.com/open?id=1WPsVhtBQmdgQl25_evlGQ1mmTQ...
    - mlechha 8y ago
      Atiyah says, Hirzebruch named and defined the function. Why not look for it in the work of Hirzebruch that Atiyah cites at the end?
  - wbl 8y ago
    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.
  - m00n 8y ago
    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.
- FredrikMeyer 8y ago
  I haven't watched the Abel lecture yet. What is it that you find bizarre?
ginnungagap 8y ago
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.
pg_bot 8y ago
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 8y ago
  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_formula_for_the_Riemann_zeta_function 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.
  - guicho271828 8y ago
    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?
deleted 8y ago
[deleted]
deleted 8y ago
[deleted]
ssmmww 8y ago
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 8y ago
  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).
JohannFlobuster 8y ago
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 8y ago
  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.
- MichaelBurge 8y ago
  If RH is independent, then it's true - since any specific counterexample has a concrete algorithm & proof to locate it.
blurrrrr 8y ago
Here’s the reference [2]. https://drive.google.com/file/d/1WNbTDKljpUR-4im-IxqluY1tKer88-mH/view https://drive.google.com/file/d/1WNbTDKljpUR-4im-IxqluY1tKer...
danbruc 8y ago
PDF version of the paper. https://drive.google.com/file/d/17NBICP6OcUSucrXKNWvzLmrQpfUrEKuY/view https://drive.google.com/file/d/17NBICP6OcUSucrXKNWvzLmrQpfU...
ramshorns 8y ago
Is there a version of this in English somewhere? I can't read Chinese.
- aaaaaaaaaab 8y ago
  Scroll down.
pvitz 8y ago
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/17NBICP6OcUSucrXKNWvzLmrQpfUrEKuY/view https://drive.google.com/file/d/17NBICP6OcUSucrXKNWvzLmrQpfU...
orbifold 8y ago
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
blinom 8y ago
I proved RH more that a year ago over the proof of the convexity of square of modul of zeta function on x on critical strip see my arxiv this article is true up to some corrections in calculuc tomorrow will appear almost final version mr Atjah many times appear in the google search of pictures blinosky prove riemann hypothesis What happen nexr there starting the philosophy that Newton stands on the shouldes og giants or something else my wife afraid fo my helth ir widely announced even if I win there is serios danger for my real life. Vladimir Blinovsky We are rather young and whant tolive
- JoshuaAshton 8y ago
  What is grammar?
  - blinom 8y ago
    Here you are right
    - blinom 8y ago
      He devoted preprint to his wife - the same as me :(
- fusiongyro 8y ago
  There is some discussion of your paper on math.stackexchange here: https://math.stackexchange.com/a/1900048/24124 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.
  - blinom 8y ago
    You are true - but it is not Hollywood believe me - it is real life, I will repeat better I proved RH more that year ago, it is in my arxiv, https://arxiv.org/abs/1703.03827up https://arxiv.org/abs/1703.03827up to some corrections in calculus ((almost) final version appear tomorrow) it is right proof. I proved the convexity of the square of modulus of zeta function over x on critical trip. Mr Atiyah many times appear in the in the pictures google search "Blinovsky proved Riemann hypothesis". What will be next - if his proof is wrong (I dot check it) - nothing - if it is also valid - started philosophy that Newton stands on the shoulder of Gians, or something else. Even if I "win", I will not be O, but I will be happy to live next 25 years with my wife Blinovsky Vladimir ps. He devoted his preprint to his wife the same as me :(
sygnon 8y ago
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 8y ago
  > 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.
LifeInSlowLane 8y ago
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 8y ago
  That's what I've heard. They're letting it slide to not make a big deal of it in media.

99 points by chenzhekl 8y ago

I’m guessing this is related to this? https://www.newscientist.com/article/2180406-famed-mathematician-claims-proof-of-160-year-old-riemann-hypothesis/ 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-atiyahs-conference-on-the-riemann-hypothesis https://mathoverflow.net/questions/311062/sir-michael-atiyah...

letitgo12345 8y ago

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

reikonomusha 8y ago

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

tanderson92 8y ago
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.
mlechha 8y ago
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.
- vlasev 8y ago
  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_evlGQ1mmTQE0Ww4a https://drive.google.com/open?id=1WPsVhtBQmdgQl25_evlGQ1mmTQ...
  - mlechha 8y ago
    Atiyah says, Hirzebruch named and defined the function. Why not look for it in the work of Hirzebruch that Atiyah cites at the end?
- wbl 8y ago
  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.
- m00n 8y ago
  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.
FredrikMeyer 8y ago
I haven't watched the Abel lecture yet. What is it that you find bizarre?

ginnungagap 8y ago

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.

pg_bot 8y ago

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 8y ago
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_formula_for_the_Riemann_zeta_function 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.
- guicho271828 8y ago
  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?

deleted 8y ago

[deleted]

deleted 8y ago

[deleted]

ssmmww 8y ago

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 8y ago
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).

JohannFlobuster 8y ago

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 8y ago
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.
MichaelBurge 8y ago
If RH is independent, then it's true - since any specific counterexample has a concrete algorithm & proof to locate it.

blurrrrr 8y ago

Here’s the reference [2]. https://drive.google.com/file/d/1WNbTDKljpUR-4im-IxqluY1tKer88-mH/view https://drive.google.com/file/d/1WNbTDKljpUR-4im-IxqluY1tKer...

danbruc 8y ago

PDF version of the paper. https://drive.google.com/file/d/17NBICP6OcUSucrXKNWvzLmrQpfUrEKuY/view https://drive.google.com/file/d/17NBICP6OcUSucrXKNWvzLmrQpfU...

ramshorns 8y ago

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

aaaaaaaaaab 8y ago
Scroll down.

pvitz 8y ago

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/17NBICP6OcUSucrXKNWvzLmrQpfUrEKuY/view https://drive.google.com/file/d/17NBICP6OcUSucrXKNWvzLmrQpfU...

orbifold 8y ago

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

blinom 8y ago

I proved RH more that a year ago over the proof of the convexity of square of modul of zeta function on x on critical strip see my arxiv this article is true up to some corrections in calculuc tomorrow will appear almost final version mr Atjah many times appear in the google search of pictures blinosky prove riemann hypothesis What happen nexr there starting the philosophy that Newton stands on the shouldes og giants or something else my wife afraid fo my helth ir widely announced even if I win there is serios danger for my real life. Vladimir Blinovsky We are rather young and whant tolive

JoshuaAshton 8y ago
What is grammar?
- blinom 8y ago
  Here you are right
  - blinom 8y ago
    He devoted preprint to his wife - the same as me :(
fusiongyro 8y ago
There is some discussion of your paper on math.stackexchange here: https://math.stackexchange.com/a/1900048/24124 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.
- blinom 8y ago
  You are true - but it is not Hollywood believe me - it is real life, I will repeat better I proved RH more that year ago, it is in my arxiv, https://arxiv.org/abs/1703.03827up https://arxiv.org/abs/1703.03827up to some corrections in calculus ((almost) final version appear tomorrow) it is right proof. I proved the convexity of the square of modulus of zeta function over x on critical trip. Mr Atiyah many times appear in the in the pictures google search "Blinovsky proved Riemann hypothesis". What will be next - if his proof is wrong (I dot check it) - nothing - if it is also valid - started philosophy that Newton stands on the shoulder of Gians, or something else. Even if I "win", I will not be O, but I will be happy to live next 25 years with my wife Blinovsky Vladimir ps. He devoted his preprint to his wife the same as me :(

sygnon 8y ago

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 8y ago
> 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.

LifeInSlowLane 8y ago

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 8y ago
That's what I've heard. They're letting it slide to not make a big deal of it in media.