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Prominent Mathematicians Rebuke Recent Riemann Hypothesis Proof
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."
I guess they mean that there's no shame in having failed, since many other respectable attempts also failed.
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.
Nope. We can do calculations that involve n-bodies, of which obviously 3-body is part, but they involve using the 2-body solution of Newton for all unique pairs in a simulation.
A separate general three body solution probably does exist, but no-ones found it.
If found, it would quite possibly revolutionise n-body modelling, and prove useful to space science (if, and only if, it sped up calculations), but I doubt astronomers would care much.
A learning experience is one of those things that say, 'You know that thing you just did? Don't do that.' - D. Adams
I believe you're mixing this up with another hard problem that hasn't been proven yet. You're thinking about the NP = P 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)
Just wanted to point out that Professor Connes is also a Fields medalist (1982).
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.
The function you remember is f(x)=1 for rationals and 0 else. It isn't possible to integrate this function as a Riemann integral, but it is still Lebesgue integrable. If f(x)=1 only for integers then the function is integrable also in the Riemann sense.
I am a mathematician, and there's no reference for this claim, but RH is a problem in analytic number theory and none of the credible work on it (meaning not by random crackpots) uses anything involving factoring. Why would an algorithm to factor numbers have any use at all, especially since this isn't something that can be proven computationally anyway?
The best we've done algorithmically by assuming the Riemann hypothesis is come up with faster algorithms to test primality (like an unconditional Miller-Rabin algorithm) or better bounds on runtime (as in "PRIMES is in P"), but these use properties of the primes that shed absolutely no light on how to factor composite numbers. Other consequences of the Riemann hypothesis tend to be things like tighter bounds on the prime counting function, and these are analytic estimates which again don't say anything useful about factoring. Determining discrete information like the prime factors of a given integer just doesn't ever seem to come out of it.
Li did respectable work once and has made a large faux pas in his handling of this affair, but it is now over. Let's focus on something far more interesting if we're talking about the Riemann Hypothesis - a wonderful (translation of a) transcript of an interview with Atle Selberg, which makes fascinating reading.
Addressing the latter first, Newton's equations describe to a very high degree of accuracy (perfectly, in the limit of ignoring relativistic and other high-order corrections) the interaction of any arbitrarily large number of bodies. The fact that we cannot solve these equations is in no way a failure of the models - the only possible failure is if we found them to be incorrect in some way. Provided they continue to produce correct results (as can be verified by two-body experiments and extended to n-body through numerical modelling, if nothing else) then the models are correct. That they are hard (or impossible) to solve in general has no bearing on the validity of the model - it tells us how they work, the fact it doesn't fit neatly into analytic mathematics is an irrelevance to how the universe proceeds.
With regards to the nature of proof in physics as opposed to mathematics - it is not generally correct to say that a "proof" of a physical theory has been found, but rather that its predictions have been verified against experimental evidence. A (correct) mathematical proof is by definition irrefutable: proving the Riemann Hypothesis would mean it is true, with no dispute. On the other hand, every bit of evidence supporting Newtonian mechanics, relativity, or any other physical theory is only valid until an exception appears, and then the theories must be updated, leading to a series of increasingly exacting tests.
The recent "proofs" of theories which have been around for decades are really only these more stringent tests - and as applications typically require orders of magnitude less precision than the level required to test a theory at a given time, it is unsurprising that theories can be easily be applied to these much less difficult test cases.
Something like the Riemann hypothesis is quite interesting, as it falls somewhere between the two - there is a certain degree of "experimental mathematics", if you will, where people are valiantly trying to find the limits of the hypothesis, which thus far indicates that it holds for a very wide range of numbers, which is comparable to the tests physicists must perform to attempt to determine physical laws. These results are encouraging as they validate any proofs which only put similar requirements on the hypothesis, but there is a higher level of proof in mathematics which would verify it in all conditions, everywhere, which would in turn validate all theories based on the hypothesis completely, and close the loophole that they will break in some (obviously ill-defined) conditions.
(Also, as an addendum, I assume you meant Poincaré spent a long time on the n-body problem, as opposed to Pasteur who was more of a biologist, as far as I recall)
Are you sure about that? Getting a paper onto arxiv.org doesn't seem to be that hard, and there's lots of ways to find out about it (RSS feed, etc.). He may not have had any reason to believe that he'd get this sort of attention, as he may have thought everyone involved would simply assume that it wasn't worth much, not having been peer reviewed.
While I love the free and open flow of information that arxiv represents, this is hardly the first time that something has been posted on there and subsequently blown out of proportion. The Internet at large doesn't seem to really understand arxiv.org, that just because someone's got a fancy LaTeX paper up claiming some wild thing doesn't mean it's credible. A paper on arxiv.org shouldn't even be understood as being endorsed by the author, let alone "science". I always love when somebody backs up their argument about physics with a link to arxiv.org, it's like a red flag that it's time to just pack it in, you're not going to get through to this person, because they only understand the trappings of science, not the actual process.
That's a meaningless statement for several reasons:
1. Problems in P and NP return either true or false on each input, so "determine the prime factorization of N" is not really in either class. AFAIK it's unknown whether any sufficiently close problems of that form are in P, are NP-complete, etc.
2. More importantly, "P=NP" is a conjecture about problems solved by Turing machines. Quantum computers are not equivalent to Turing machines (they're too strong), so having a polynomial-time algorithm for some problem on a quantum computer gives us no information about whether or not it's in P.
Not quite. A set of measure zero is not necessarily empty. For example, the set of rational numbers is measure zero inside the reals. See here. Also, 'place' is a technical term. See here for a definition.
This post expresses my opinion, not that of my employer. And yes, IAAL.
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
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
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
My english is sow-sow. Sowhat?
Inductive is a philosophical term, the inference of new facts based on previously known ones. In Physics, this means using experimental data in order to make general assumptions about the universe.
In mathematics, we use the term tongue-in-cheek, to refer to a particular and useful consequence of the least-element axiom. It resembles inductive reasoning, but it is indeed quite more rigorous.