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Prominent Mathematicians Rebuke Recent Riemann Hypothesis Proof
Posted by
ScuttleMonkey
on Fri Jul 04, 2008 02:29 PM
from the here-is-your-peer-review dept.
from the here-is-your-peer-review dept.
Bryan writes "Xian-Jin Li's purported proof of the Riemann Hypothesis (reported on recently) has been rebuked by Fields Medalist Terence Tao. Fortunately, Dr. Li's proof fails alongside a respectable graveyard of previous attempts." Relatedly, jim.shilliday writes "The proof cites and appears to be based in part on the work of the leading French theorist Alain Connes. A few hours ago, Connes posted a comment on his blog stating that the purported proof is so badly flawed that he stopped reading it."
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Claimed Proof of Riemann Hypothesis 345 comments
An anonymous reader writes "Xian-Jin Li claims to have proven the Riemann hypothesis in this preprint on the arXiv." We've mentioned recent advances in the search for a proof but if true, I'm told this is important stuff. Me, I use math to write dirty words on my calculator.
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Why "fortunately"? (Score:5, Interesting)
From the summary:
Fortunately, Dr. Li's proof fails alongside a respectable graveyard of previous attempts
Why? I'm probably missing something obvious, I'm not even a mathematician to start with, but...
I mean, we (the world) do want to prove it right (or wrong) one day or another, don't we?
Re:Why "fortunately"? (Score:5, Informative)
I guess they mean that there's no shame in having failed, since many other respectable attempts also failed.
Parent
Re:Why "fortunately"? (Score:5, Insightful)
They're just being polite by pointing out there's no shame in failing to prove the Riemann Hypothesis, since it has frustrated the attempts of many a prominent mathematician so far.
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Re:Why "fortunately"? (Score:5, Insightful)
Having said that, Li is no crank. I had not heard of him, but that's no surprise since I'm not a number theorist. But he has published several refereed papers in this area, has a position at BYU, and really ought to have known better than to explode on the scene like this.
I've gotten communications from genuine crackpots, wanting my comments on their work. Early in my career, I wrote back, gently pointing out the mistake. To my horror, friends then received slightly modified but still absurd drafts, listing me as a collaborator! Li is a real mathematician, probably with poor social skills, and a bad proof.
Parent
Re:Why "fortunately"? (Score:5, Informative)
I believe you're mixing this up with another hard problem that hasn't been proven yet. You're thinking about the NP = P [wikipedia.org] problem. The difference is that here we don't know what will be the outcome, whereas for the RH most people assume it's true. Having a proof for this wouldn't really change anything (apart from validating large parts of mathematics that assume it is true)
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Re:Why "fortunately"? (Score:5, Funny)
[..] lead to the undermining of current encryption methods, which depend on the difficulty of factoring large prime numbers.
That's a trivial problem.
All prime numbers have two factors: 1 and itself.
Goodbye encryption :-)
Parent
Re:Why "fortunately"? (Score:5, Informative)
One possible explanation for your understanding (which in my understanding, is wrong), is the Miller-Rabin primality test algorithm.
The primality problem (telling whether a number is prime), although hard, was never proved to be NP-complete.
The Miller-Rabin primality test is a (actually, the 1st and possibly the only) polynomial deterministic algorithm that is based on the Riemann hypothesis (polinomial deterministic meaning "fast and accurate"). Proving RH would prove that Miller-Rabin is exact and therefore shown that primality testing is in P.
http://en.wikipedia.org/wiki/Miller-Rabin_primality_test [wikipedia.org]
Unfortunately, algorithm freaks were faster than math freaks (well, the algorithm freaks involved were math freaks too) and a new algorithm called AKS was developed that did everything Miller-Rabin did without relying on the Riemann Hypothesis.
http://en.wikipedia.org/wiki/AKS_primality_test [wikipedia.org]
So, to this day, we know primality testing is polynomial. The _real_ problem in cryptography is prime *factoring* (if it's not prime, then find 2 numbers that when multiplied produce the original number). Although it is not know whether that problem is P or NP-complete or both, it is believed to be outside NP because it is much harder than plain primality testing.
http://en.wikipedia.org/wiki/Integer_factorization [wikipedia.org]
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I don't know about you all... (Score:5, Funny)
The "proof" is that of Theorem 7.3 page 29 in Li's paper, but I stopped reading it when I saw that he is extending the test function h from ideles to adeles by 0 outside ideles and then using Fourier transform (see page 31). This cannot work and ideles form a set of measure 0 inside adeles (unlike what happens when one only deals with finitely many places).
Re:I don't know about you all... (Score:5, Insightful)
Use the "Star Trek" filter:
he is extending the test function h from [ tech ] to [ tech2 ] by [ tech 3 ] and then using Fourier transform ... This cannot work and [ tech ] form a set of measure 0 [ tech 4 ] (unlike what happens when one only deals with finitely many places).
When he moved from one set to another and did the Fourier transform, he forgot that he ended up with an empty set instead of a finite number of points because that's apparently a property of whatever the hell he was talking about.
Parent
Re:I don't know about you all... (Score:5, Insightful)
It's called "Proof by Intimidation":
using the formula:
[ some formula ]
it is trivial to see that:
[ some other formula out of nowhere ]
therefore, combining the above, we can arrive at the easily obtained answer:
[ some MATLAB result ]
Don't forget, it works both ways; the people marking your assignment don't want to admit that they can't see the so-called "trivial" derivation.
Parent
Ow my head (Score:5, Funny)
The proof, and the rebuke, only proved my theory that there is a distinct surge in advil usage when something like this is posted on /. or digged.
Preprint, not a reviewed paper (Score:5, Insightful)
Well duh this is what we have been saying - this is a preprint and is likely to have errors. Whether or not they can be repaired is open to question.
Wiles' proof of Fermat's last theorem took a long time to go through the review and repair process. And there was at least one pretty hard problem that had to be fixed.
Slashdot's "journalistic" process really suxors when it comes to this sort of stuff.
Re:Preprint, not a reviewed paper (Score:5, Funny)
Slashdot's "journalistic" process really suxors when it comes to this sort of stuff.
Wel of course it does. Slashdot is journalology, not journalonomy.
Parent
Lazy title selection (Score:5, Funny)
"Renowned Researchers Rebuke Recent Riemann Reasoning"
What a shock... (Score:5, Interesting)
Re:What a shock... (Score:5, Interesting)
I work part-time for a couple of mathematics research journals and we do get the occasional crank submission. There's one guy who's been sending us, on average, a 'paper' every week or so for the past few years: typically a single, badly-written page of gibberish (we're talking Time Cube [wikipedia.org] standard lunacy here) which is clearly not the work of someone who's ever seen a real mathematics paper. We've never responded to him, or even acknowledged any of his submissions (helpfully he prints his return address on the back of the envelope, so these days they go straight in the bin, unopened and unread) and yet he still keeps sending them in.
The arXiv also tends to get its fair share of crank submissions, usually elementary attempted (but trivially broken) proofs of things like the Goldbach Conjecture, Fermat's Last Theorem and the like - I'm assuming that the really mad stuff is filtered out by the moderators.
In contrast, at a quick glance to my nonspecialist eyes (I'm a knot theorist) Xian-Jin Li's preprint looks like a genuine (if flawed) attempt by a serious, qualified mathematician who specialises in the relevant area. Fair play to him for trying, though. I'm also not sure I'd characterise Terence Tao or Alain Connes' refutations as 'rebukes' - they looked more like dispassionate analyses of the paper's flaws to me, the sort of discussion you'd expect from the peer-refereeing process.
Parent
Prof Connes also a Fields medalist (Score:5, Informative)
Just wanted to point out that Professor Connes is also a Fields medalist (1982) [wikipedia.org].
I guess it is a testament to Xian-Jin Li excellent reputation and the importance of the topic that these two mathematical superstars took the time to look at his proof.
there was no rebuke (Score:5, Insightful)
The proof was a legitimate effort by a non-crackpot, but the ideas in it were well known to specialists in the field and were generally understood to not be powerful enough to crack the problem. So the errors were found fairly quickly. Scott Aaronson's post Ten Signs that a claimed mathematical breakthrough is wrong [scottaaronson.com] item #10 may be helpful in understanding what happened.
Re:what does it all mean, Basil? (Score:5, Funny)
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Re:what does it all mean, Basil? (Score:5, Informative)
There are a lot of results based on assuming the conjecture is true, including a variety of factoring and root finding algorithms that are computationally very useful.
Until it is proven you really don't know if these algorithms are giving correct answers.
This is why it is so important and has a big prize associated to it.
Parent
Re:what does it all mean, Basil? (Score:4, Interesting)
Since the work based on the assumption that the hypothesis is true is in itself valuable, it will still be used.
It's just that a proof, if found, will elevate who-ever finds it to the status of mathematical superstar.
Consider this, we are still finding proof of various of Einstein's theories, but work based on his has been of real value for decades.
Here's another example that makes me sound all clever because I know it.
Newtons equations, and his entire body of work, completely failed to explain how it is that the moon can orbit the earth while the earth orbits the sun, and we *still* don't have the equation to explain that bugger.
There are specific n-body solutions, I've written one myself, but a solution for the general case? Nope, never been done.
Louis Pasteur spent most of his life on that particular problem, as have many other prominent scientists, all to no avail. We found a use for Newtons work regardless, and Einstein extended it successfully, even with that glaring hole.
Parent
Re:what does it all mean, Basil? (Score:5, Insightful)
You are mixing the basic tenants of physics and mathematics, not a good thing to do. Science is a mix of inductive and deductive logic, math has a higher standard and doesn't admit inductive proofs.
Three guys were vacationing in Scotland. One was an astronomer, one a physicist and one a mathematician.
In their travels they chance on a black sheep grazing in a field.
Astronomer: All sheep in Scotland are black
Physicist: Some sheep in Scotland are black
Mathematician: There is one sheep in a field in Scotland that is black on at least one side.
Parent
Re:Not Making Yourself Look Good Here (Score:5, Insightful)
Parent
Re:Not Making Yourself Look Good Here (Score:5, Insightful)
Yes, why don't you tell the Fields medalist how to make himself look good? I'm sure he needs your help desperately. Jeebus, you know that a Fields medal is objectively harder to get than a damned Nobel prize, right?
He did critique the 'proof' objectively. The claim was that by looking at the function on a certain domain ("ideles" whatever those are), one could look out from there and see how it would have to behave elsewhere ("adeles"). However, the "ideles" aren't big enough to give a good viewpoint of what's going on (i.e. the function at the ideles is not necessarily representative of the rest of the function). If you only look at multiples of 2pi, you could "prove" that sin(x)==0. Just because you or I couldn't notice the obvious problem in the RH proof, doesn't mean that it doesn't merit quick dismissal. Sometimes obvious mistakes are made in math (some would say that only obvious mistakes are made - but they are only obvious once they are pointed out).
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Re:Not Making Yourself Look Good Here (Score:5, Insightful)
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