Claimed proof of Riemann Hypothesis | Hacker News Reader

Claimed proof of Riemann Hypothesis

researchgate.net

66 pointsjapaget27 comments

Is there someone here who follows Riemann Hypothesis research closely enough to comment on whether there is any there here?[1] The Riemann Hypothesis is a sufficiently complicated and famous problem that I think it must be easy for a wishful thinker to suppose he has found a solution when he actually hasn't.

[1] This is a reference to a famous quotation from Gertrude Stein's autobiography, "anyway what was the use of my having come from Oakland it was not natural to have come from there yes write about it if I like or anything if I like but not there, there is no there there."

https://en.wikipedia.org/wiki/Gertrude_Stein#.22There_is_no_...

That quote makes me feel like I'm having a stroke.

On page 4, it says "Now by Robin criterion d(n)<0 for n large enough, yielding lim sup (n->infinity) (d(n)) <= 0". If I understand the criterion from page 1 right, assuming RH is false only implies that d(n) <= 0 for some n > 7! -- not for all n sufficiently large, so the limit superior is not constrained as claimed. In fact, on page 2, the paper claims "Thus, if m is bounded and n->infinity, we see that d(n)->infinity", which, if the falsity of RH did imply lim sup (n->infinity) (d(n)) <= 0, would make for an even shorter and simpler proof.

Disclaimer: I've only got an undergrad in math and don't know much about the specifics of the cited papers, so I might be missing something.

Apparently the negation of RH is enough to state that there needs to be infinitely many numbers such that d(n)<=0. That would imply that you can find arbitrarily large n such that d(n)<=0, but I don't think that's enough to claim that "lim sup (n->infinity) (d(n)) <= 0".

I think it's enough to claim "lim inf (n->infinity) (d(n)) <= 0" though, but for some reason they proved that in a different way, that I don't quite understand yet.

There are dozens of papers attempting to prove the Riemann Hypothesis. Here is a list [1]. A joke from the author of the list: "It's easier to prove the RH than to get someone to read your proof".

[1] http://empslocal.ex.ac.uk/people/staff/mrwatkin//zeta/RHproo...

Coming from a PhD in math I can give this good trick for assessing grand mathematical claims:

Google the authors.

Maybe unfair to intelligent amateurs, but based on my decade of experience you find out from this whether to take something seriously.

Might need some adjustment of Google terms for hard-to-google names, just use common sense.

Except Patrick Sole is a real mathematician who has been working in this area for some years now http://www.emis.de/journals/JTNB/2007-2/article03.pdf and so it's not easy to dismiss it just based on a name.

The most plausible explanation is https://www.reddit.com/r/math/comments/3vnrqj/two_authors_cl... here: "Zhu sent Sole some questions about his Robin inequality paper, including Zhu's ideas for proving RH. Sole responded, but there was some communication breakdown that led to Zhu thinking Sole endorsed his ideas. Zhu typed up his idea and added Sole's name to it in order to get the paper read. This is of course unethical, but given that Zhu thought his proof was correct, in his mind he was doing Sole a favor."

This was published on Saturday so my best guess is Patrick Sole on Monday will either post a refute or will claim it is true and everyone will shit a brick (unikely).

I wasn't clear enough. I was responding to the flow of comments of the form: "Riemann hypothesis is hard, this is unlikely to be true." Sure it's true, but doing a little more research could inform that opinion well past the zeroth-order approximation of "it's a hard problem."

I didn't actually take my own advice, I just wait for Terence Tao to write a post then I know it's true :)

I think the average HNer will hear when a proof happens for real in just the everyday channels of theirs: Twitter, Facebook, Reddit, HN etc will be FULL of it (with good reason). Remember the Higgs boson?

The description on researchgate says "work in progresswork in progresswork in progresswork in progresswork in progresswork in progresswork in progresswork in progresswork in progress". It seems unlikely that the author would write that if they really believed this was a complete work. It's just not worth posting purported proofs of the Riemann Hypothesis. There are dozens of them. Until someone serious comes forward and says, "we think this is a proof", it's not.

I'm just going to say "no way". No way that a 4-page document is the proof, and no way it's on any site other than a university or Arxiv.

The paper addresses that directly:

"Thus, while our proof of RH might look too short to be true, it really involves at least two rather long papers [6, 8], not to mention [9]."

Strange, that sentence has already been rewritten.

I am surprised that in 30 years since [8] no one took another step toward the result in this new paper.

Arxiv detracts from, does not add, to credibility. it's an open preprint platform after all. This is like saying "No way such an important project would be anywhere other than github." That's not much of an argument.

As for flaws, it's four pages. They can be written up in an afternoon, if the paper makes sense to its intended audience. Of course, that's not exactly HN.

EDIT: Very interesting replies. I'll leave this up, though apparently quite wrong.

Arxiv is not for all,it is only authors vouched by other members. It is a very weak form of peer review.

Arxiv is pretty much for all. What you call vouching is just endorsement, and it's not peer review at all.

what I got Googling 'pseudoscience arxiv' - http://arxiv.org/find/all/1/all:+AND+marvin+herndon/0/1/0/al...

These titles wouldn't even make it past HN's front page, they would be flagged - by lay people with no background.

Let me emphasize: arxiv is not peer-reviewed, it is an open publication platform.

Out of the above list, I only clicked through one - http://arxiv.org/abs/physics/0510090 reading "Teaching Earth Dynamics: What's Wrong with Plate Tectonics Theory?"

Quote:

"I review what is wrong with plate tectonics theory ... and describe my new Whole-Earth Decompression Dynamics Theory, which unifies the two previous, dominant theories in a self-consistent manner. Along the way, I disclose details of what real science is all about, details all too often absent in textbooks and classroom discussions."

This wouldn't make it past a HN reader. "I disclose details of what real science is all about." No peer would ever accept that, in any journal on any subject.

There are vast areas of research where every credible paper gets posted to the arxiv. So even though being on the arxiv does not imply credibility, not being on the arxiv implies lack thereof.

I'm incredulous of:

>There are vast areas of research where every credible paper gets posted to the arxiv

due to your word "every" which from context you mean literally; no exceptions. So, what would these vast areas of research be? The statement would not be true for any areas of research I know about.

EDIT: Thanks for the replies - fascinating. That in certain fields, no paper is worth reading if not on arxiv.

For the areas I used to work in for many years, high energy physics and quantum gravity, I can't think of a single exception.

Of course I don't have total knowledge of these fields, only my corner of them, and I'm sure you'll find a counterexample if you dig deep enough, but if counterexamples are remarkable and rare, then the heuristic: "Not on the arxiv, not likely to be credible" stands.

High-energy physics. Yes, literally every paper that's worth your time is on arxiv first. The only time anyone reads from an actual peer-reviewed journal is when something that was on arxiv eventually gets published (you know, a year later) and might have minor corrections. And I don't know a single person that picks up journals for those crucial papers that were never posted to arxiv.

What's the point of "formal peer-review", anyway? If you're in the community you talk to your colleagues about what's on arxiv and you know what's good and what isn't. You don't need a journal to tell you.

I will add that astrophysics is the same way. I'm not aware of a single paper I've come across from the last 15 years that isn't on the arxiv.

I can vouch for this. Lots of theoretical physics on there (in fact, I would say the majority of it). Whether stuff is posted on arXiv highly depends on the field.

I would say that yes, if a project is on github (or any other openly accessible platform) it greatly adds to the credibility of said project. Just like papers that are early-released on arvix it allows me to form an opinion about it and more importantly allows more intelligent people than me to read the code and draw conclusions which I can then lean on.

So in other words...what's the downside of publishing on arvix? You can also send it to a peer reviewed journal.

One thing many people don't understand is that sometimes it is easier to prove a stronger result than a particular result, which could be exactly this case. I don't know if this paper is correct, but the technique used seems plausible independent of the length of the paper.

Any errors in the actual content of the paper shouldn't take long to turn up. There seems to be nothing more high-powered than calculus used in the proofs.

At the end of Lemma 2, "[9, Th. 8, (39)]" seems to be referring to corollary 1 of Theorem 8 on page 70 of [1], equation (3.30). Maybe "(39)" is meant to be "(30)".

Their argument for Theorem 1 seems not-crazy, and quite accessible.

[1] https://projecteuclid.org/download/pdf_1/euclid.ijm/12556318...

The paper uses results from "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann" by Robin, which is not available online, as far as I can tell.

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