# A Has the Riemann hypothesis been proven?

• Featured

#### martinbn

Science Advisor
My guess is that he is writing crackpot papers on purpose, as an experiment. Checking if people will take them seriously, or at least give them some attention, just because they come from a famous mathematician.

#### Auto-Didact

For those who still wanna see it:

#### fresh_42

Mentor
2018 Award
My guess is that he is writing crackpot papers on purpose, as an experiment. Checking if people will take them seriously, or at least give them some attention, just because they come from a famous mathematician.
This would at least be in the best tradition of English humor, but I seriously doubt it.

#### Auto-Didact

My guess is that he is writing crackpot papers on purpose, as an experiment. Checking if people will take them seriously, or at least give them some attention, just because they come from a famous mathematician.
After having watched his lecture and read most of his preprint (The Fine Structure Constant), I'm convinced the man is dead serious. I'm not particularly fond of the manner in which many younger people (read: mathematicians, students and just a while bunch of random people on the internet) seem to be patronizing him.

Even if everything Atiyah claims regarding the RH is false, they probably still aren't fit to untie his sandals; there is a reason you don't see the likes of Tao and Schulze making such remarks about the man for they understand that sometimes silence can be golden.

#### fresh_42

Mentor
2018 Award
... there is a reason you don't see the likes of Tao and Schulze making such remarks ...
So true. I find it far more interesting to discuss, why Polson (Chicago), Stenger (Salt Lake City) or Blinovsky (Moscow) aren't discussed, although all of them published a proof on arxiv.org recently - and all of them are mathematicians.

#### nrqed

Science Advisor
Homework Helper
Gold Member
Bullshit! Everyone should be scrutinized for their work and not for who they are.
I apologize if I have insulted people by putting weight on the fact that the claim of a proof had been made by someone who has earned both a Fields medal and an Abel prize. Seriously, I understand... I recall being annoyed by someone who used to post thousands of posts discussing more the affiliations of authors of papers, who they had worked with, who their supervisors were, where they had done their postdocs and on and on, than their actual work.

#### Auto-Didact

I apologize if I have insulted people by putting weight on the fact that the claim of a proof had been made by someone who has earned both a Fields medal and an Abel prize.
We need to put weight on the claims of such people in an intellectual community such as academia; what else are these prizes good for if not to publicize the towering proven intellect of these remarkable individuals? It is no coincidence that although tonnes of people, many of them even extremely skilled experts, work in mathematics and physics today, not just anyone of the experts is or can be regarded as a Newton, a Gauss or a von Neumann. This can be encapsulated in the difference between being capable of inventing calculus in the 1600s by yourself with no clear precedent and merely being able to do calculus, after being spoonfed a rigorous theory of calculus in undergraduate mathematics courses.

Moreover, this doesn't seem to be that well of a known fact among scientists and mathematicians generally (there are notable exceptions), but there is even a striking statistical demonstration (NB: first discovered empirically in the social sciences (!) and then generalized mathematically) which justifies this argument, namely that for any valuable skill(set) which one can be better at then some other and the results of which are generally valued by others, there exists a distribution such that the most valued results produced by all practitioners of such a skill is disproportionately produced by a small subset of the entire population of practitioners; among that small subpopulation of skilled people the same thing holds again i.e. an even smaller subset in approximately the same proportion is again responsible for the production of the large majority of the most valued results.

What this means in this discussion is that there are some scientific works that are much more read than others, generally indicating their superior perceived value, and a small number of works which practically everyone has read. For those who already do know this, they will recognize immediately that I am speaking about none other than the Zipf-Pareto principle which can be described by a very simple power law and/or further mathematicized into a very special kind of probability distribution; what most people (probably) do not yet know is that there is even an elegant piece of pure mathematics underlying the scale-invariant self-similarity of this ubiquitously occuring distribution, which ties together the mathematics underlying probability theory, modern network theory, fractal geometry and (nonlinear) dynamical systems theory among others, but I digress.

To get back to my point, Sir Michael Atiyah is exactly the towering kind of intellect, that has shaped not just the physics and mathematics of his time but an entire generation of thinkers probably in more ways than they can or indeed do realize; Edward Witten for heaven's sake is directly among the man's mathematical offspring. To even try and compare yourself, let alone put your mind above his, would mean that you are not merely some celebrated expert in a particular field of mathematics such as algebraic geometry, but simultaneously an expert in mathematical physics, having contributed to countless related mathematical fields and having almost 70 years of experience of being an expert and letting all that knowledge and experience shape his thoughts; just try and let that thought sink in for a moment.

Mathematicians like Atiyah are a class apart from pedestrians such as you and me, who are literally runts trying to mimic the gods themselves; although the gods may be fallible, so much more can we be. Not being able to recognize the limits of one's own intellect is a very common fault and feature of those not lucky enough to be counted as part of the pantheon (yet). The only living public figures in science I can even think of who are somewhat properly comparable to Atiyah, and I say this with very much a reserved judgement, are themselves lone stellar intellects, namely Roger Penrose and Gerard 't Hooft; anyone who knows anything about the average scientists' perception of these two distinguished gentlemen will fully understand that it is the shame of our generation, as it is of those before them, that we do not venerate our heroes more during their life.

It is in this respect that especially scientists can still learn an awful lot (both good and bad) from the general public, i.e. ordinary citizens of the world: publically celebrating the birthdays of our living heroes en masse for example wherein we celebrate both their life and work, instead of only suddenly finding the inspiration to publically appreciate their life and work when their death is announced, while in the meantime pushing nonsensical trends such as Pi Day in a hopeless effort to try and connect with the public; a real and honest public display of affection from scientists for their own heroes would do very much for the public appreciation and dissemination of science. To end on a positive note, here is a piece about another mathematical giant, written posthumously by Atiyah: a tribute to Hermann Weyl. I just hope that others will show the same kind of care and respect for Atiyah, not just after he has gone and left us, but more importantly while he is still with us.

Last edited:

#### MTd2

Gold Member
I think he is making bold statements because he doesn't think there is much time ahead for him. So, he is pushing the ball and claiming a proof so that people may think more about the direction he is pointing to. He is using a type of language and inspiration that is more keen to Physicists. I don't think he is expecting much from Mathematicians, as there is not enough time for such formalities.

#### Auto-Didact

There were two google drive documents shared that were supposedly Atiyah's work. Here someone used it to calculate the fine-structure constant, and the result is horribly wrong.
I read the first paper where the equations are from. This entire reddit thread is a bit disingenuous or rather quite misleading to say the least, because they use Eq 1.1 and 7.1 to perform the calculation, while Atiyah clearly states that to calculate $\alpha$ the equations in section 8 are required. Now admittedly, the text is difficult to penetrate... however, be that as it may, that in no way justifies carrying out a strawman calculation and then declaring the whole thing to then be wrong.

The explicit series is explained in section 8, specifically 8.1 through 8.6, while the actual explicit function is given in 8.11 based on some Bernoulli polynomial in 8.10; I agree that the presentation of the series given here is a bit opaque, but having reread the entire thing a second time certainly helps, especially after having listened to the talk with slides.

This infinite series is, in contrast to the more familiar infinite sums and infinite products, an infinite exponentiation, i.e. something of the form $2^2^2^2^2^...$. I've definitely seen iterated exponents before but I am simply not that familiar with infinitely iterated exponents and under what conditions and circumstances they can be said to converge in general or not. In either case, Atiyah claims something about the whole thing being convergent if 8.7 and 8.8 are arbitrarily close, with fixed $t ^m$ given $t$ is sufficiently small.

Atiyah tries to explain it himself a bit further in the text:
page 13 said:
We can describe what we are doing in the following way. Given any number $2^n$, we can factor it as a product of two numbers $2^{n(0)}2^{n(1)}$where $n = n(0) + n(1)$. As $n$ gets larger, we keep $n(0)$ fixed, say $n(0) = 4$, and let $n(1)$ get larger. This describes our chosen algorithm and explains the shift by 4 with $t(n) = v(n + 4)$. This will give the correct 12 digits. When we increase n, to improve on the approximations $Ж(n)$ we will have to increase $n(0)$ and $n(1)$, but we cannot be sure of their optimal values. However, since our sequences are monotonic increasing, we can adopt the stopping rule : stop one step before the product $(8.7)$ exceeds the sum $(8.8)$. This can be formalized in terms of the Bernoulli numbers $B^n_k$ of higher order which, as explained below, are essentially Hirzebruch’s Todd polynomials.

#### mfb

Mentor
while Atiyah clearly states that to calculate $\alpha$ the equations in section 8 are required.
That is a contradiction. We get two unambiguous formulas, one to calculate "ch" and one to calculate the fine-structure constant based on "ch". Why would you need anything else if the formulas were correct?

If I ask you to find x, and tell you that 2+5=x, do you need to read section 8 of my post to find x? Section 8 might have a different way to do so (in this case it is unclear what section 8 actually suggests to do), but surely it should give the same result.

#### Auto-Didact

That is a contradiction. We get two unambiguous formulas, one to calculate "ch" and one to calculate the fine-structure constant based on "ch". Why would you need anything else if the formulas were correct?

If I ask you to find x, and tell you that 2+5=x, do you need to read section 8 of my post to find x? Section 8 might have a different way to do so (in this case it is unclear what section 8 actually suggests to do), but surely it should give the same result.
This actually isn't true if one formula given requires initial conditions given by another formula, i.e. if the formula is somehow stated incompletely. The situation worsens considerably if we equate analytical formulas with approximative numerical formulas, without carrying out the approximation numerics correctly.

This is exactly what is stated here, in the bolded parts:
pg 8 said:
To use (7.1) for computation, we need to specify the initial data, something which will be done in section 8. The numerical verification that Ж agrees with 1/α to all decimal places, so far calculated, follows from the numerics of section 8. This comes in three steps, the first involving the sum and integral of the formulae (1.1) and (7.1) as with γ. But, as Euler discovered, the convergence in this process is too slow for effective computation.
A computer will only give the correct results given a sufficient speed of convergence.

Now the situation even gets hairier in section 8 when he starts using iterated maps of exponentials. I think the way the section is written in that multiple backtracks are necessary is what creates much of the ambiguity and ensuing confusion. Especially the statement regarding ignoring he first term in 8.5, when comparing 8.5 and 8.6 seems to trigger distrust by most readers that something must be amiss.; mathematically speaking however, the argument is clear using induction.

Further confusion then seems to arise again for 8.7 and 8.8 (especially for those not very
familiar with using the monotone convergence theorem and/or limit comparison test) because he backtracks and then talks about, I'm presuming, an unfamiliar technique to most readers. This gets worse because he then uses the Mars rocket analogy instead of a mathematical argument and I suppose most people, definitely pure mathematicians, just give up reading even though he gives a reference to Hirzebruch’s proper demonstration in the very next section even explaining how $t$ needs to be interpreted differently, how the stopping rule is justified because of monotone convergence and how to formalize this using Bernoulli numbers.

Another historical backtrack to Eddington throws the reader off again before he finally ends by giving an explicit prediction in 8.9 based on his usage of the formula in section 8.

It seems to me that this backtracking and throwing in of historical sidenotes in the main text is the main problem with his paper for most readers, especially his throwing in of theological metaphors and the word 'magic': his style of writing is blatantly non-Bourbakian and therefore suggestive of being "not proper formal mathematics"; it instead reeks of popular science writing. Many others and myself will agree that his writing style is definitely non-Bourbakian, but this has no bearing whatsoever on his mathematical argument itself; just dismissing some argument because you don't like how it is written is definitely a case of throwing out the baby with the bathwater.

For the younger people who do not know this: the Bourbakian writing style characteristic of contemporary academic mathematics is a very novel invention, which only became universally standard in the mathematical community long after the generation of Atiyah were already working mathematicians.

Important to note is that physicists don't use it, and many old mathematicians, especially those that also do physics, actually chose not to adopt the Bourbakian writing style, because it is that and that only: a writing style. Of course, used correctly, it can be much more clear than regular writing but that is only because it is overly pedantic, while being simultaneously absolutely sterile in a literary sense.

Last edited:

#### mfb

Mentor
Sorry to be direct now but this is nonsense. The formula is simply wrong. There is nothing incomplete about a simple equation.
The situation worsens considerably if we equate analytical formulas with approximative numerical formulas, without carrying out the approximation numerics correctly.
Unlike the publication claims (that is another error), the series converges quickly. In addition all the terms after the first few are negative - the partial sums are always larger than the limit, but they are too small to produce the fine structure constant.
Now the situation even gets hairier in section 8
It does not matter. 2+5=x in the real numbers defines x in a clear, unambiguous way no matter how much you write elsewhere about what you want. Same for equation 7.1.

What that wrong formula means for the rest of the publication is a different question. If there is a part that is both easy to understand and easy to check that is completely wrong I don't have much hope for the part that is difficult to understand and hard to check.

#### Auto-Didact

Sorry to be direct now but this is nonsense. The formula is simply wrong. There is nothing incomplete about a simple equation.
By that argument alot of simple formulas are "wrong". Example: $E = mc^2$ and $E = P/t$, therefore, $mc^2 = P/t$.
Extremely simple, I wouldn't even put it pass a high school kid to use such an argument, but what is wrong here? Even though simple it should be clear that the above is obvious nonsense if used because of missing context and multiple implicit simplifications in both of the equations. I'll respond to the rest later.

#### mfb

Mentor
Your examples relate physical properties of objects in specific systems to each other. Equation 7.1 doesn't do that, it is a purely mathematical equation.

By the way: I'm not sure what E=P/t is supposed to represent.

#### Auto-Didact

Your examples relate physical properties of objects in specific systems to each other. Equation 7.1 doesn't do that, it is a purely mathematical equation.
For 7.1, at least the RHS of the equation, that might be true but the point is I'm not so sure that that is even true here for the LHS as well or for 1.1 for that matter.

All that seems to be given is that $T(\pi) = Ж$ and $T(\gamma)=Ч$, which makes me immediately conclude that 1.1 is either a simplification i.e. taken as is algebraically incorrect. Without an explicit definition of T nothing further can be said, and that is why we need a definition of T, which is given in section 8; also I would start out by saying that whatever T is it is definitely not an analytic function.
By the way: I'm not sure what E=P/t is supposed to represent.
That was my entire point: essential context is missing!

#### mfb

Mentor
For 7.1, at least the RHS of the equation, that might be true but the point is I'm not so sure that that is even true here for the LHS as well or for 1.1 for that matter.
Well, the two sides are equal. The right side is just a well-defined real number so the left side has to be a well-defined real number as well. There is no context necessary for a real number. This is different from your example where you used tons of undefined variables.
which makes me immediately conclude that 1.1 is either a simplification i.e. taken as is algebraically incorrect.
If it is algebraic incorrect and therefore incorrect why is it in the paper? Anyway, that's what I am saying: 7.1 / 1.1 are incorrect.

#### Auto-Didact

Well, the two sides are equal. The right side is just a well-defined real number so the left side has to be a well-defined real number as well. There is no context necessary for a real number.
Again, there is no guarantee that $T(\pi) = Ж$ and $T(\gamma)=Ч$ are real numbers, or even numbers for that matter; they could be sets of numbers, strange kinds of maps themselves, weird hyperfunctions, physical quantities, you name it.
This is different from your example where you used tons of undefined variables.If it is algebraic incorrect and therefore incorrect why is it in the paper? Anyway, that's what I am saying: 7.1 / 1.1 are incorrect.
I was in a hurry and made it up on the spot, we could extend or change the example, but I think I already made my point clearly enough without needing to resort to examples: context dominates in physics, the symbols have a meaning, you can't just go around equating any quantity just because they happen to have a symbol in common. Even if two symbols represent the same general quantity it still may be completely inappropriate to directly equate them, especially if you leave out subscripts, and expect to get an answer which isn't complete nonsense. Moreover having prior knowledge, which is literally knowing the context beforehand, enables unpacking a simplified equation if necessary such as is possible with $E=m$ in multiple ways.

As for why 1.1 is in the paper, its safe to say the preprint wasn't checked by anyone else. Its either just an error or perhaps some kind of abuse of notation or shorthand, meaning something like 'the relation of $Ч$ to $\gamma$ is the same (or analogous) to the relation of $Ж$ to $\pi$'.
This doesn't immediately invalidate the entire rest of the paper, that would be potentially throwing out the baby with the bath water.

#### martinbn

Science Advisor
Again, there is no guarantee that T(π)=ЖT(\pi) = Ж and T(γ)=ЧT(\gamma)=Ч are real numbers, or even numbers for that matter; they could be sets of numbers, strange kinds of maps themselves, weird hyperfunctions, physical quantities, you name it.
By the definition of $T$ they have to be complex numbers. The definition of $T$ itself seems confused to me.

#### Auto-Didact

Small development over at https://math.stackexchange.com/questions/2930742/what-is-the-todds-function-in-atiyahs-paper
The entire thread is interesting, but the most interesting part is that someone decided to email Atiyah asking about the Todd function:
Jose Brox said:
I just decided to email Atiyah asking for clarifications, and he has answered. If I figure something worthy out of the conversation, I will post it here (of course, since I'm not an expert in analysis, I may fail to understand subtle ideas). For starters, the preprints are from him (although he didn't know they had leaked, and is going to circulate a paper), and address the "T would be constant" issue: since it is defined as a weak limit (which is not unique), it has no analytic continuation. It is uniquely determined by Hirzebruch theory. If you want to help me, write to josebrox at mat.uc.pt – Jose Brox 8 hours ago
By the definition of $T$ they have to be complex numbers. The definition of $T$ itself seems confused to me.
In the same thread this was posted, referring to page 122: http://120.27.100.167/uploads/soft/all/18729.pdf

Last edited:

### Want to reply to this thread?

"Has the Riemann hypothesis been proven?"

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving