Riemann Hypothesis Proved? 454
Theodore Logan writes "Has the Riemann Hypothesis finally been proved? The proof is a couple of months old, and to the best of my knowledge a Swedish newspaper is the only one to take up the story yet, so there is certainly a possibility that this is a hoax, or a less than watertight proof. But if it turns out to be the real thing, it will, apart from winning the authors eternal fame and glory for finding the holy Grail of modern math, provide them with a cool $1 million as they claim the first Millennium Prize." We had a story a while back about this as well.
Okay, assuming this proof to be correct... (Score:1, Informative)
Wow ... (Score:3, Informative)
-- shayborg
Don't get too excited yet... (Score:5, Informative)
Re:um... (Score:5, Informative)
-- shayborg
Attempt at putting it in more layman's terms. (Score:5, Informative)
We are going to show you beyond a shadow of a doubt that the non-trivial zeros of the zeta-function are of the form 1/2 +- i*theta_n.
It is based on the construction of an infinite family of operators D^{(k,l)} in one dimension, and their respective eigenfunctions \psi_s (t), parameterized by continuous real indexes k and l.
To do this, we are going to use the operators D^{(k,1)} and their respective vectors \psi_s (t), such that using D^{(k,1)} on \psi_s (t) will produce k*(\psi_s (t)), where k is some non-zero constant. Unfortunately though, we have to show a way to product all of these operators. So the "construction of" the operators will be contained within the proof.
Orthogonality of the eigenfunctions is connected to the zeros of the Riemann zeta-function.
These \psi_s (t) vectors are also all at "right-angles" to eachother. So their cross products = 0.
Due to the fundamental Gauss-Jacobi relation and the Riemann fundamental relation Z (s') = Z (1-s'), one can show that there is a direct concatenation among the following symmetries, t goes to 1/t, s goes to \beta - s (\beta a real), and s' goes to 1 - s', which establishes a one-to-one correspondence between the label s of one orthogonal state to a unique vacuum state, and a zero s' of the \zeta.
Z(s') = Z(1-s') is true. Thus, we can show that there is a connection between the follwing symmetries:
t goes to 1/t,
s goes to \beta -s (where beta is a real number),
and s' goes to 1 - s'
In Q.M. we can show then a correspondence between one of these orthogonal states to a unique vacuum state (from Quantum Mechanics), and thus a solution of the zeta function.
It is shown that the RH is a direct consequence of these symmetries, by arguing in particular that an exclusion of a continuum of the zeros of the Riemann zeta function results in the discrete set of the zeros located at the points s_n = 1/2 + i \lambda_n in the complex plane.
From these neat little tricks, we can show that the Riemann Hypothesis must be true, because these things are true.
Statement of the hypothesis (Score:5, Informative)
The hypothesis states that all (nontrivial) zeroes of the zeta function occur on the line Re(z) = 1/2.
If proved, it has immense implications in many areas of pure and applied mathematics. For instance, in number theory: it would say a lot about the distribution of prime numbers.
The stature of the problem can be seen from the fact that it was one of the 23 problems which would shape the mathematical progress of the 20th century that David Hilbert drew up in his lecture at the 1900 Paris congress of mathematicians.
From a Swede (Score:4, Informative)
Slashdot them to hell. It's my university, they can take it.
HINT: Go read the comments on the previous article (Score:5, Informative)
by njj (133128) on Tuesday July 02, @12:05PM (#3808279)
(http://www.csv.warwick.ac.uk/~marem/
If you can't explain something in ordinary words to a layman, then you really don't understand it.
I'm about halfway through writing up my PhD thesis on some applications of homological algebra to knot theory and low-dimensional geometric topology (provisional title liber rerum dementiae, but it'll probably end up being called something more mathematically appropriate).
In principle, yes, I could explain the details of my research to a suitably motivated layman. But I suspect it would take rather a long time.
You see, and this really isn't meant to sound arrogant, supercilious, or dismissive, university-level mathematics is pretty damned difficult, and the details of most cutting-edge research really doesn't make sense until you've spent several years learning the background (the mindset, the language, the fundamental concepts).
My current area of research is essentially the applications of homological algebra to knot theory and low-dimensional geometric topology. To explain this to a non-mathematician, I'd first have to teach them a lot of background stuff (group theory, a bit of stuff about rings and modules, point set topology, basic algebraic topology (the fundamental group, (co)homology theory), some geometric topology (basic course in knot theory, some stuff about 3-manifolds), a bit of category theory, and some homological algebra (broad overview of the (co)homology theory of groups and algebras)).
It's taken me nearly nine years (3-year BA, 1-year MSc specialising in topology and knot theory, plus nearly five years doing a (part-time) PhD) to get to this point myself. If I were a bit cleverer (or didn't have a `proper' job as well) I might have been able to shave a couple of years off that.
My friend Steve has a physics degree. I managed, in ten minutes one evening, with much handwaving, to give him some idea of what my thesis is all about. It helped that he knew what a group was already though. But for me to explain it fully to him would probably necessitate him doing at least one mathematics degree first. And that's not really something I'd wish on one of my friends
Now this really isn't meant in an arrogant way, and I hope you won't read it like that, but Euclid was right: There is no royal road to geometry.
I can have a go at explaining the Riemann hypothesis, though. To fully understand what it's about and why it's so damned difficult you'll need to do an advanced course in complex analysis (which isn't my field either).
A complex number is a sort of two-dimensional number, which you can regard as a point in a plane (the `complex plane' or `argand diagram'). You add them together coordinate-wise, and you multiply them together in a weird manner which involves something which behaves like a `square root of -1' (engineers also like to think of it as a sort of 90-degree phase-shift operator, I'm told).
There's a particular function (`Riemann's zeta function') defined on the complex plane (it takes one complex number as input and returns one complex number). For some complex numbers (`the zeros of the function'), the value of this function is zero.
The `trivial' zeros occur at the points -2, -4, -6,
The `non-trivial' zeros (that is, all the other points for which zeta is zero) all seem to occur on the line parallel to the vertical axis that intersects the horizontal axis at +0.5. Indeed, nobody's ever found one which doesn't.
The Riemann Hypothesis is that *all* the non-trivial zeros lie on this line. It's known to be true for the first (large number which temporarily escapes me), but it turns out to be phenomenally difficult to prove that it's true in every case.
Now that's the basic idea, but it doesn't (and I can't - it's not my field) explain *why* it's so difficult that some of the greatest minds (Hardy, Littlewood, Ramanujan, etc) of the past 150 years have failed to prove it, and why the Clay institute are willing to pay a million dollars to someone who can.
- nicholas (we don't just sit around doing big sums, you know
Re:Okay, assuming this proof to be correct... (Score:5, Informative)
This proof has already gone down in flames (Score:5, Informative)
I know the editors of this site mean well, but what we have here is a link to a site that defines the Riemann Hypothesis in very abstract terms, a link to a LANL preprint from two completely unknown researchers deposited there in November 2002, and a link to an obscure Swedish newspaper from almost two weeks ago, and no other supporting material. So my BS meter is running at 5.
The odds that "this is the one!" given that pedigree would seem to be really tiny. But the clincher for me is the following web page dedicated to would-be proofs of the Riemnann Hypothesis [ex.ac.uk] whose important text is (and I quote):
And the Castro and Mahecha preprint (and another grandiosely titled preprint by Mahecha) is linked to from there. Now my BS meter is running at about 9. So now I check for messages abou this at deja.com in the sci.math group. [google.com] Read the thread yourself; it's pretty entertaining.
So, with my BS meter running at 11, the work having been submitted for coming up on 6 months, and no indication whatsoever that this is real, I suggest it is false.
And I also suggest that Slashdot might wish to consider contacting a real mathematician to filter their potential stories on mathematics, since I can't tell you the last time one of these "is X finally proven?" stories has panned out.
An Explaination of what this means (Score:5, Informative)
Re:Grammer different? (Score:3, Informative)
"Usage Note: Prove has two past participles: proved and proven. Proved is the older form. Proven is a variant. The Middle English spellings of prove included preven, a form that died out in England but survived in Scotland, and the past participle proven, a form that probably rose by analogy with verbs like weave, woven and cleave, cloven. Proven was originally used in Scottish legal contexts, such as The jury ruled that the charges were not proven. In the 20th century, proven has made inroads into the territory once dominated by proved, so that now the two forms compete on equal footing as participles. However, when used as an adjective before a noun, proven is now the more common word: a proven talent."
Go figure.
KFG
Re:Attempt at putting it in more layman's terms. (Score:2, Informative)
I mean dot product... Sorry about that.
(for those of the unitiatied)
dot product means
A . B = sum(a_n*b_n), for all n.
cross product is something completely different.
Re:um... (Score:4, Informative)
It has been years since I studied this, and even then I didn't fully understood it. So if there are any mathematicians reading this out there, please feel free to correct any misconceptions I might have.
Reimans hypothesis basically states that there is a correleation between the distribution of prime numbers (how many numbers are in between each of them) and a complex function (complex in the sense that it deals with imaginary numbers). In theory, you could use this complex function to predict the space there is between one prime number and the next one. Therefore you could use this function to predict which would be the next prime number given any other.
One possible field of appliction is encryption, which strongly relies on the mapping of this numbers.
Human Translation (Score:5, Informative)
Here's a human translation:
Classical Math Problem May Be Solved
One of the great unsolved problems of mathematics, the so called Riemann Hypothesis, may have a solution 144 years after Bernhard Riemann published his idea of a special equation, related to prime numbers.
If the proof does turn out to be correct, the two problem solvers may be eventually be able to collect a one million dollar reward.
In the year 1900, the world's leading mathematicians gathered for a conference in Paris. During the conference, David Hilbert, the leading mind of mathematics at the time, presented 23 problems which would affect mathematics for the ensuing century, and yet today. One of these problems was the Riemann hypothesis, and despite great effort it has remained unsolved. However, in November of 2002, Carlos Castro of Clark Atlanta University, Atlanta, USA, and Jorge Mahecha of the University of Antioquia, Medellin, Colombia published a proposed solution.
They won't be able to collect the one million dollar reward offered by the american Clay Mathematics Institute until one year after publication. This is to allow other mathematicians time to check the result, and verify its correctness.
- If this is true, then it is a sensational. This is a class of problems which requires much work before a possible solution can be confirmed. Thousands of mathematicians all over the world will cast themselves at this, and examine the evidence in minute detail, says Anders Karlqvist, mathematician, professor of informatics and head of the Polar Research Secretariat.
He explains that certain of Hilberts problems are problem areas rather than individual problems. Some have also been reformulated. If the Riemann Hypothesis is solved, then ten of the 23 problems are solved, seven are unsolved, five are partially solved, and one is no longer relevant.
According to Anders Karlqvist, Hilbert was a great man within his field, and a great period ended with him. He was the last man to have an overview of all the fields of mathematics.
Mathematics have developed very rapidly during the latest decades, thanks to an aid that the mathematicians of the early 20th century couldn't predict - the computer. The ever faster and larger computers of today can handle vast quantities of numbers and quickly make calculations that used to be impossible for a person even if he or she spent an entire lifetime.
With the help of computers, certain problems have been solved, such as the four color problem. It says that at most four different colors are neccessary to colour a map, so that areas with a common border don't have the same color. A computer program has systematically gone through all possible combinations.
Anders Karlqvist thinks that this involves a philosophical dilemma: should proofs in the form of computer programs be accepted? He believes that we stand before a culture shift within mathematics. During the coming decade mathematics will develop radically due to the increasingly efficient computers.
Bengt Jonsson
Re:This proof has already gone down in flames (Score:5, Informative)
1. SvD isn't an "obscure" Swedish newspaper. It's the biggest, counting readers in if not millions so at least hundreds of thousands. They seldom print bogus.
2. That the proof hasn't been verified yet doesn't mean it can't be correct.
3. The sci.math discussion doesn't really say anything about the validity of the proof, only that, as you say, the paper seems to not have been proof-read very well, etc.
But, I agree that in essence you have reasonable complaints. BS meter at 11 is quite high, though. Mine is at about 5. 11 is reserved for make-money-fast schemes and herbal viagra.
Oh, and one more thing. An AI posted a translation of the article [slashdot.org] that seems to have gone largely unnoticed. (Just in case there is someone in here who isn't fluent in Swedish.)
Re:You can help (Score:3, Informative)
Sure you might find a counterexample instead of a proof. But all the OP said was that you wouldn't find a proof.
Re:This proof has already gone down in flames (Score:4, Informative)
While I have no doubt that SvD is of singular value to almost everybody (especially for its largest entries on the diagonal), the fact is that it comes from Sweden makes it orthogonal to our concerns. (Sorry about that...)
Here in the US, you have to understand that unless you share a border with Iraq, we just don't have time to be interested in you these days. :-)
True enough, but see below, and the fact that if *I* had a proof of the Riemann Hypothesis, I probably wouldn't submit it to the high energy physics and "general math" sections of xarchiv.
Actually, the discussion basically says that nobody could read the thing and that it was chock full of typos. Once again, if I had a proof of the Riemann Hypothesis, I would probably make very certain that it was free of such interest-busting material. A mathematician can feel free to correct me, but I am assuming that while raw papers (especially from non-native speakers of the language the paper is submitted in) can be tough to get through, you usually *do* detect the high quality of the real work at some point pretty early on, and if you don't, you assume the worst.
In the thread I referenced, a physicist chimed in with the observation that one of the co-authors (Castro) was not taken seriously in his own field of physics, which makes it even less plausible that the math in this paper would be new and inspiriational.
Unfortunately, the Clay Math Prize has kind of made proving the Riemann Hypothesis a "make-money-fast" scheme. :-)
Or to put it another way, here's the plan of attack I believe the authors had:
Re:Breaking Encryption? (Score:3, Informative)
Quantum encryption, IMO, is the answer - if you even *try* to decrypt it, you muck it up and both the sender and receiver know about it.
Millenium Prize description for RIemann's Hypo... (Score:2, Informative)
z(s) = 0
lie on a straight line. This has been checked for the first 1,500,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers.
> Can anyone explain what they mean by "interesting solution"?
Re:Attempt at putting it in more layman's terms. (Score:2, Informative)
Re:No one noticed this? (Score:5, Informative)
For example, in the _very first_ equation, he introduces an "operator", and conveniently forgets to mention what space this operator is supposed toact on. A Banach space? A Hilbert Space? We should not have to _guess_ what algebraic structure they're using.
YAW.
BS meter (Score:3, Informative)
Re:Why bother proving it... (Score:2, Informative)
Re:point from the swedish article (Score:2, Informative)
The skeptecism in using a computer comes up when we let a chip 'think' for us (or rather, just follow the steps). A mathematician may argue that while a human's logical argument is always sound in a formal system (let's just ignore Godel shall we) there is no guarantee that the same will be true in a computer simulation. There is no guarantee that a couple of extra electrons won't pass through some transistor giving me a 2 instead of a 1. There are a lot of things that can go wrong. We can run the program a million times and be reasonably sure we got the right answer, but never a 100% sure.
In reply to sql*kitten:
Computers can be used to prove infinite cases, if the problem is approached in the right way. Look up the proof of the Four-Color Theorem [wolfram.com] to see what I mean.
Re:um... (Score:2, Informative)
Now physics, on the other hand, and the other physical sciences, you could get away with proof by experimental evidence. But mathematicians are a strange breed, haha.
Re:In that case -- Yay! I win a million dollars! (Score:4, Informative)
I.e., s, which is a complex number, has two parts - a 'real' part and an 'imaginary' part. Thus, z(s) for any complex s returns zero - according to this proof - if s.r (the 'real' part) and s.i (the 'imaginary' part) lie along a certain straight line.
Make more sense now?
Re:HINT: Go read the comments on the previous arti (Score:2, Informative)
Such is the lot of the mathematically literate. People and software lose attention and label you as lame...
Lemme see if a short summary can get through though. The Riemann zeta function is Z(x) = 1/1^x + 1/2^x + 1/3^x +
That made the Riemann Hypothesis interesting. But nobody knew how to solve it. So they put in work figuring out what it would prove, and what might prove it. And they found that a lot of other interesting statements that connect in some way to the distribution of primes were all equivalent to the Riemann Hypothesis. Every one of these both improves the chance that we can prove it and makes the problem more frustrating. (Remember, for mathematicians frustrating = interesting. Yeah, math types are weird.)
Computer folks might recognize this pattern - the same thing happened with the P=NP conjecture. It is what happens with any famous math problem.
And what is the estimate of the distribution of the primes? Well consider the function isPrime(n) that returns 1 if n is prime and 0 otherwise. It isn't a random function, obviously, but in a lot of ways isPrime(n) looks like a random function that has probability 1/log(n) of returning 1. (That is the natural log.) Try it. Write that function, and feed in a bunch of random integers around a million. You will find that for every 100 you put in, about 7.2 of them are prime.
Given that, you could guess that the number of primes less than or equal to N should be roughly:
1/log(2) + 1/log(3) +
And roughly should be (if you remember your statistics...) off by something random and generally proportional to (N/log(N))^0.5. (Basically the errors that you would expect from a random walk - the 1/log(n) is because the probability of a 1 is 1/log(N).)
Well that is the estimate. What do I mean by saying that that estimate is really good? I don't actually mean as good as suggested above (which is how good it looks in practice), I mean o(n^(0.5+epsilon) for every epsilon bigger than 0. In other words it might not be quite sqrt(N), but it looks kinda like that...
So basically, the Riemann Hypothesis would confirm a lot of things that we suspect about the distribution of prime numbers. It would also prove a lot of other things that we came up with while trying to find a proof for the Riemann Hypothesis.
I would say more detailed things, but that involves facts and Slashdot thinks that those are lame.
Its very short... (Score:2, Informative)
cf. Fermat's Last Theorem proof that's tiny.
I suppose we'll see, its been submitted to a journal, and will be undergoing peer review, and lots of checking.
Here is what the hypothesis says, clear and crisp: (Score:2, Informative)
Let's first define what 'root' of an 'equation' is. Let's say our expression is
3x + 6
to find the roots, we say this equation is equal to zero, and solve for x:
3x + 6 = 0
thus x = 2. So, the only root of the equation is the real number 2. There are also complex numbers - which are written as
a + bi
where a and b are the numbers you already know, the real numbers, and the symbol 'i' donates the square root of -1. a is called the real part of the complex number, and bi is called the imaginary part of the complex number. So let's say,
3.2 + 4i
is a complex number. And the 'real part' of this complex number is 3.2 Now, equations, polynomials etc. often have roots that are complex numbers.
We use 5^2 to mean five squared, and in general x^y means x to the power of y. Consider this expression:
1 + 1/(2^s) + 1/(3^s) + 1/(4^s)
This is an addition of infinite terms. This can also be expressed easier in summa-notation. The above is called the zeta function. Now we have enough background to state the riemann's hypothesis:
Riemann's Hypothesis
--------------------
The real part of all the complex roots of the zeta function is 1/2.
For example, the following complex number is a root that satisfies the hypothesis:
1/2 + 14.1347i
I hope this makes it a lot clearer. Mathematicians have checked up to some billionth root numerically, but a proof means a thoroughly logical, very tight argument that is valid for all numbers, which nobody 'officially' has yet.
Now, the relation others wrote about crypto is mainly crap. It is true that this is closely related to prime numbers, and it is related to crypto, but the other stuff people wrote on this forum is mainly fabrication. Existence of a proof or otherwise won't change the current crypto-systems at all. A good rule of thumb is not to believe in journalists, or college students who are not much better.
Savasa Hayir.
Istanbul / Turkey.