Has the Riemann hypothesis been proven?

  • #51
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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.
 
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  • #52
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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.
 
  • #53
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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.
 
  • #54
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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.
 
  • #55
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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!
 
  • #56
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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.
 
  • #58
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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.
 
  • #59
martinbn
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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.
 
  • #60
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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
 
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  • #64
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Here are 3 current proofs of RH/GRH published on arxiv.org beside Sir Atiyah's.
https://www.physicsforums.com/threa...thesis-been-proven.955832/page-1#post-6061194
This only shows, that it is obviously a vital area of research. Whether one of them will actually do the job hasn't been decided as of now.

They are not part of this discussion, so please do not promote them (referring to a removed post).
It would appear in this case, part of getting credit for the proof, for whoever eventually gets credit for it, will include for the person being able to acquire enough of an audience, that there will be at least a couple of people who study the proof in enough detail to verify it.
 
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  • #66
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According to my newspaper Atiyah said he didn't really want to go public just yet.
 
  • #67
A heated debate, dont count out the old man yet.This could be a gift for all of us.

personally , i believe mortality drives people to do plenty of things, his closeness death most likely stimulated his genius.

thats said lets wait to see the proof.
 
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  • #68
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This. Reading the paper carefully instead of brashfully shows that there seems much to be gained which just might not have been expressed very precisely, analogous to when one confuses a Lie group G for its Lie algebra ##\mathfrak {g}##. These kinds of errors are made very frequently and typically aren't any real cause for alarm.

These kinds of errors, which are similar to forgetting some process during a routine larger process such as seasoning during cooking, are the types of mistakes older people easily tend to make while the rest of their mental faculties are still very much intact. Given Atiyah's age and his therefore possibly (if not likely) slowly deterioting mental condition, it is no wonder he is making such cavalier mistakes, which are are easily spotted and correctable by experts.

Non-experts, especially unexperienced youngsters including new assistant professors, postdocs and lower tend not to be capable of understanding such subtleties because they haven't worked yet or long enough in (academic) practice for years on end for them to have developed such an intuition. If they see such a mistake they tend to take it literally and then altogether dismiss the rest of the work as probably unsalvageable without giving it any due diligence.

To refer back to my earlier analogy, if your grandpa who was once a Michelin star chef forgets to put some seasoning in the food during the process of preparing a grand feast meal for the entire family and then goes on to serve the meal, upon tasting that there is something off, you don't just throw away all the food he prepared and then mockingly question your grandpa on his ability to cook; instead you just add some seasonings.
 
  • #69
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These kinds of errors, which are similar to forgetting some process during a routine larger process such as seasoning during cooking, are the types of mistakes older people easily tend to make while the rest of their mental faculties are still very much intact. Given Atiyah's age and his therefore possibly (if not likely) slowly deterioting mental condition, it is no wonder he is making such cavalier mistakes, which are are easily spotted and correctable by experts.

Non-experts, especially unexperienced youngsters including new assistant professors, postdocs and lower tend not to be capable of understanding such subtleties because they haven't worked yet or long enough in (academic) practice for years on end for them to have developed such an intuition. If they see such a mistake they tend to take it literally and then altogether dismiss the rest of the work as probably unsalvageable without giving it any due diligence.
That really encapsulates the dark irony of scientific and mathematical research, doesn't it? Either you're too young to understand the subtleties or you're too old to remember why they're important. It must leave like six months out of your entire life where you're capable of being fully productive :/
 
  • #70
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That really encapsulates the dark irony of scientific and mathematical research, doesn't it? Either you're too young to understand the subtleties or you're too old to remember why they're important. It must leave like six months out of your entire life where you're capable of being fully productive :/
This actually seems to apply to practically all professions in which experts frequently can and need to employ subtle reasoning, not just science and mathematics. The situation in mathematics is just far more opaque, for most even almost wholly reliant upon the actual deferral of reasoning about the matter to a small group of other people, which hopefully are experts in the matter at hand.

The issue is therefore far more susceptible to subjective bias than in other fields, unless those few to whom the reasoning is deferred are actually willing to fairly i.e. objectively give an argument its due diligence. This situation is exactly analogous to the situation in law and medicine, except that in those fields there are dire consequences for the small group of experts involved if it can be shown that the experts just chose to be negligent out of convenience.
 
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  • #71
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I have finished reading the paper for the third time and in doing so I have noticed a very curious coincidence: at the end of one of my earlier posts in this thread, post #47, I linked to a biographical memoir written by Atiyah about Hermann Weyl I had come across a few years ago when I was reading up on Weyl. In it, on page 328, Atiyah says the following about Weyl:
Michael Atiyah said:
Weyl was a strong believer in the overall unity of mathematics, not only across sub-disciplines but also across generations. For him the best of the past was not forgotten, but was subsumed and refined by the mathematics of the present. His book The Classical Groups was written to bring out this historical continuity. He had been criticized in his work on representation theory for ignoring the great classical subject of invariant theory that had so preoccupied algebraists in the nineteenth century. The search for invariants, algebraic formulae that had an intrinsic geometric meaning, had ground to a halt when David Hilbert as a young man had proved that there was always a finite set of basic invariants. Weyl as a disciple of Hilbert viewed this as killing the subject as traditionally understood. On the other hand he wanted to show how classical invariant theory should now be viewed in the light of modern algebra. The Classical Groups is his answer, where he skilfully combines old and new in a rich texture that has to be read and re-read many times. It is not a linear book with a beginning, middle, and end. It is more like an elaborate painting that has to be studied from different angles and in different lights. It is the despair of the student and the delight of the professor.
These ideas of the unity of mathematics,
historical continuity and especially the non-linear nature of a text which has to be read and reread again many times in order to be properly understood seem to be eerily reflected in the way Atiyah's preprint 'The Fine Structure Constant' was written; on the face of it, the numbered paragraph format is also somewhat reminiscent of Wittgenstein's Tractatus Logico-Philosophicus.

Did Atiyah write the paper this way on purpose, knowing it would probably only be understandable by the older readers? As I have argued in my earlier posts including #68 in this thread, much of the controversy seems to stem from the way this paper is written. I haven't tracked down Weyl's book yet, so this remains speculation. In either case, more and more, it seems to be the case that emulating this style was exactly his intent.

For example, in my first and second reading of the paper, both times I thought his remarks about the Axiom of Choice in 6.6 were clearly erroneous and that he was confusing the axiom with the school of Brouwerian intuitionism and its rejection of the law of the excluded middle; upon my third reading however I decided to read up on the historical matter regarding the axiom of choice a bit more and learned that I just wasn't aware that the law of the excluded middle is directly derivable from the axiom of choice. In other words, during a third careful reread I realized it was in fact I who was mistaken about something based on my prior knowledge of some fact being incomplete and therefore incorrect, while he was correct all along!

As for the faulty equations, especially 1.1 and possibly 7.1 as well, it seems very clear that these bits were written later than the other parts of the text as they seemingly come from thin air. With regard to 7.1, where does this equation come from exactly if not derived from the equations in section 8? I'm beginning to fear that these bits were written (much) later than most of the other parts, perhaps after his wife had already passed or after his cognitive decline had begun/worsened, and that perhaps there are even mistakes lurking in 7.1 which are extremely difficult to even identify, let alone correct without explicitly rederiving such an expression based on the equations in section 8.
https://news.ycombinator.com/item?id=18054890

Above is a comment on his "proof".
Regarding the third comment there, quoted here for convenience here:
m00n said:
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.
Most of these points are actually rebutted by Lipton & Regan to which @martinbn linked to in post #62. Here again we see that professionals and experts have a very different grasp of matters compared to non-experts.

Moreover, I tracked down Hirzebruch's book which was referenced in the paper, in particular chapter 3. This chapter is a mere 23pp read instead of 250pp. I will see what can be found in it. If anyone wants a link to the chapter I will provide it.
 
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  • #72
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The first comment suggested that being 89 years old makes Sir Atiyah's claim less credible. I would like to believe that one's math insights steadily improve and that while age may slow the brain, it does not make one less insightful.
 
  • #73
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I have no idea what you are saying, sorry. <edit: post this refers to removed>
 
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  • #74
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I have no idea what you are saying, sorry.
Sounds like the noumenon/phenomenon distinction... I can see how the noumenon/phenomenon distinction might directly apply to bare and dressed electrons for example.

W.r.t. this thread itself however where we are talking about mathematical proof of the RH, I'd say he is attempting to say something more along the lines that mathematical structures already exist Platonically prior to their proof, i.e. it has eternally existed and will do so whether we discover it or not, just like all other extant mathematical objects.

Once such an object has been fully grasped within someones mind for the first time, that is already all the demonstration/'proof' that is necessary in his opinion. In other words, he is probably a mathematical Platonist and advocating Platonism as opposed to formalism, which has been the standard in the mathematics community since Hilbert.
 
  • #75
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Most of these points are actually rebutted by Lipton & Regan to which @martinbn linked to in post #62. Here again we see that professionals and experts have a very different grasp of matters compared to non-experts.

Moreover, I tracked down Hirzebruch's book which was referenced in the paper, in particular chapter 3. This chapter is a mere 23pp read instead of 250pp. I will see what can be found in it. If anyone wants a link to the chapter I will provide it.
In contrast to the quoted thread in post #61, Hirzebruch explains in full detail in chapter 1 (§1. Multiplicative sequences) what the Todd polynomials are. Chapter 3 goes on to expand enormously on these matters in full generality.

The first two formulae appearing on page 13 of Atiyah's paper "The Fine Structure Constant" are exact citations from Hirzebruch's book; in fact, everything stated in this paper about the Todd polynomials, Bernoulli polynormials and their generating functions can be directly traced back to this book.

It surprises me that no one seems to have taken the time to confirm this. Hirzebruch's book is actually pretty good.
 

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