Has the Riemann hypothesis been proven?

[mathman]

https://news.ycombinator.com/item?id=18054890

Above is a comment on his "proof".

 

[martinbn]

https://rjlipton.wordpress.com/2018/09/26/reading-into-atiyahs-proof/

 

[fresh_42]

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).

 

[Charles Link]

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.

 

[Vadim Sokolov]

While talking about RH, I'm probably very late to the party with this (Polson, June 2018), but does anyone know if this has received some criticism? Can't find any specifics other than the article itself.

The paper was published in a peer-review journal (https://projecteuclid.org/euclid.bjps/1528444877). Slides give an easy to absorb presentation of the work: http://faculty.chicagobooth.edu/nicholas.polson/research/polson-hilbert-8.pdf

 

[CWatters]

According to my newspaper Atiyah said he didn't really want to go public just yet.

 

[Somali_Physicist]

A heated debate, don't 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 let's wait to see the proof.

 

[Auto-Didact]

https://rjlipton.wordpress.com/2018/09/26/reading-into-atiyahs-proof/

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.

 

[jack476]

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 :/

 

[Auto-Didact]

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.

 

[Auto-Didact]

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:

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:

Spoiler

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.

 

[Ventrella]

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.

 

[mfb]

I have no idea what you are saying, sorry. <edit: post this refers to removed>

 

[Auto-Didact]

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.

 

[Auto-Didact]

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.