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More on Riemann Hypothesis
Anonymous Coward writes "The NYTimes has a little story on a recent conference at New York University's Courant Institute where mathematicians gathered to discuss potential attacks on the Riemann hypothesis. The Clay Mathematics Institute had announced an award of a million dollars for a proof (or refutation) of the Riemann hypothesis during the millenial celebrations. That million dollars won't be worth much if it takes as long as that Last Theorem by Fermat to solve. There were some interesting observations such as the statistical distribution of the zeros looked just like calculations on the energy levels of large atoms." We did a related story on hard math problems two years ago.
seeing as an NSA supercomputer could only refute the hypothesis, and out of the billions of numbers it's already tried there have been exactly zero refutations, i'd put my money on the mathematicians.
specifically, i'd place my bets on the smelliest and most Russian of them.
Cretin - a powerful and flexible CD reencoder
When they are completed, yes they are rock-solid. But in development, one tiny, almost insignificant error can throw off the whole thing.
Think about it in terms of spacecraft. A couple of vehicles were perfect and landed on Mars. One had a small defect, it wasn't complete (meters and miles were mixed up). It was lost.
I have a shitty sig!
With current technology it is extremely unlikely, although possible, that a mathematician would refute the hypothesis, but it is impossible for a computer to prove it.
While a mathematician can't try some random still untested number and hope to get the "right" one (all of the "small" numbers have been tried), he could always try and build some "special" class of numbers that could refuse the hypothesis and test those, or he could find some logical contradiction etc.
On the other side, with current technology computers can only try more and more cases, so that if the hypothesis is false they eventually find an example, but they just can't try every number (not in a finite time :) ), nor actually prove a theorem by logical means.
I know that somebody is researching some theorem-proving capable AI, but it seems that they didn't succeed yet in proving whether it can exist or not, so it will be quite some time before those could be available, if ever.
Well -- having been in this business myself, I'll try to explain the relevancy of this problem. In Fourier analysis, the problem is usually to take a look at data, apply a transform and look at the underlying harmonics of the data (this works best on sound or light because the harmonics correspond to actual physical properties such as "pitch" or "frequency"). For example such techniques can be used to clean up images by removing the "noise" type frequencies in the data. A similar technique can be applied to numeric entities such as prime numbers. Ideas such as this produce something called the Zeta function which can capture various statistical properties of prime numbers (statistical properties of the entire infinite series of prime numbers). The "zeros" of the Zeta function (I'll avoid explaining what I mean by this) capture some of the statistical properties of primes in the way Fourier transforms explain light and sound. The Riemann hypothesis comes down in essense to claiming that prime numbers never behave too statistically perverse (the equivalent in Fourier transform data of saying that the harmonics are not too skewed -- the "sound" never "gangs" up on certain frequency areas). Although the mathematics can seem a bit obscure to those outside the field, it is probably one of the more profound questions out there. As a comparison, Fermat's last theorem (when evaluated purely for its mathematical significance) is merely a curiousity and was proved as a minor consequence of proving more important theorems in elliptic curves, which themselves are only stepping stones towards another really large unproved mathematical question. For the record, the question is: "Are all elliptic curves modular?" -- which is way more obscure than the Riemann Hypothesis and I consider to be the number one potentially solvable mathematical question that has ever been posed. It is a really cool problem.
Thanks. Great explanation.
Very kind of you to say, thanks.
Could you elaborate and tie this in with the number of primes between m and n?
I'm a little less confident about this, but here goes...
As I understand it (and bear in mind that I've not done any complex analysis for several years, and number theory has never really been my forte) sometime during the 19th century Gauss noticed that the distribution of primes was approximated pretty well by a function he called the `logarithmic integral'.
Li(x) is defined as the integral from 0 to x of (1/t) dt. And apparently the number of primes below x (usually denoted pi(x)) is pretty well approximated by Li(x).
Now this is where I start to lose track of things.
As I understand it, it was proved at the beginning of the 20th century that the validity of the Riemann hypothesis is equivalent to the assertion that the deviation of Li(x) from the actual value of pi(x) is of the order of sqrt(x)*ln(x).
If I remember correctly, some work of the three great British mathematicians Hardy, Littlewood, and Hardy-Littlewood showed that pi(x) actually oscillates around Li(x) infinitely many times (although it really doesn't do it very quickly - the first value of x for which the graphs cross is very big indeed).
qv: the Clay Institute's page
and Chris Caldwell's page for better explanations.
As to whether there's a relatively simple proof out there, I don't know. My (non-specialist) suspicion is that there isn't, because some really clever people have tried to find one for a century and a half and failed. I'd be interested and impressed to be proved wrong on this, though.
Andrew Wiles' celebrated proof of Fermat's Last Theorem (if you haven't done so, read Simon Singh's book on the subject, and if possible watch the BBC Horizon documentary - the transcript is available) was pretty complicated and, I'm told (algebraic geometry isn't my field either - there's an awful lot of diverse mathematics out there) introduced some genuinely new ideas and methods.
The general feeling is that if there were a simple proof of either Fermat or Riemann (or, for that matter, the Goldbach or Poincare Conjectures) then someone would have found it by now - some really top brains have worked on all of these over the years (including several Fields medallists, FRSes, and the like).
(There's a guy who posts regularly to sci.math who reckons he's got a simple proof of Fermat which doesn't resort to all that scary stuff about elliptic curves or modular forms. The consensus seems to be that he's a nutter, though - his `proofs' contain obvious flaws which he refuses to acknowledge, claiming instead the existence of an enormous academic conspiracy against his work.)
It's also often the case (and this was true for the Fermat theorem) that proofs of such intractible problems, even those which are subtly flawed, introduce new ideas and methods of attack.
This is why otherwise sensible mathematicians have a go at these problems - even if they don't manage to solve them, the chances are that the attempt will inspire them to find new methods or potentially important partial results. Even had Wiles' original (flawed) proof turned out to be irrepairable, it was a pretty major piece of work which introduced some important new ideas which could well be useful in solving related problems.
My guess (as an interested non-specialist) is that while a proof of RH would be complicated and elegant, it would also involve some new twist or idea. As for who might do it, my money would be on Prof Louis de Branges of Purdue University - he demolished the (similarly intractible) Bieberbach Conjecture in the 1980s and thus seems to know what he's doing. Or it might be someone else entirely, someone who's spent seven years locked in their attic (as Wiles did).
nicholas