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Riemann Hypothesis Proved?
Posted by
Hemos
on Mon Mar 03, 2003 10:15 AM
from the cracking-the-problems dept.
from the cracking-the-problems dept.
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.
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Riemann Hypothesis Proved?
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Riemann hypothesis (Score:5, Funny)
um... (Score:4, Funny)
"A proof of the Riemann's hypothesis (RH) about the non-trivial zeros of the Riemann zeta-function is presented. 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. Orthogonality of the eigenfunctions is connected to the zeros of the Riemann zeta-function. 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. 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."
What the fuck?
Re:um... (Score:5, Informative)
-- shayborg
gosh, gee willakers (Score:4, Funny)
While the linked site does provide a layman's interpretation of the topic, when you first click to that page you are presented with:
Two Plus Two Equals Four
Thought we had all been trolled... :)
Re:In that case -- Yay! I win a million dollars! (Score:4, Informative)
(http://www.wherethesundontshine.net/ | Last Journal: Tuesday May 27 2003, @04:48PM)
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?
Attempt at putting it in more layman's terms. (Score:5, Informative)
(Last Journal: Saturday September 04 2004, @10:35AM)
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.
Re:Attempt at putting it in more layman's terms. (Score:5, Funny)
I was with you right up to the point where you started typing.
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.
ERROR: STACK OVERFLOW! SYSTEM POWERING DOWN...
I heard two laymen discussing this... (Score:5, Funny)
Construction Worker Joe: I think the non-trivial zeros of the zeta-function are of the form 1/2 +- i*theta_n
Construction Worker Larry: I agree. It's clear when you consider the operators D^{(k,1)} and their respective vectors \psi_s (t)
Construction Worker Joe: Of course, so long as using D^{(k,1)} on \psi_s (t) will produce k*(\psi_s (t))
Construction Worker Larry: Yeah. Joe, you'd better make sure the Eigenvectors of those two boards you're nailing together have a dot product of zero. The last time the boards weren't orthogonal, the boss had a fit!
Construction Worker Joe: Yeah, whatever. Hey, check out that girl's hyperbolic curves!
Re:I heard two laymen discussing this... (Score:4, Funny)
(http://www.certainkey.com/)
Re:Attempt at putting it in more layman's terms. (Score:5, Interesting)
(Last Journal: Saturday September 04 2004, @10:35AM)
you have an equation
f(x) = x(x-2)
now, x=0 is a trivial zero, because well anything times 0 is zero, so it's trivial, let's ignore it.
while, x=2 is a non-trivial zero, because it is unusual.
(to the mathies out there: I know, I know, this isn't 100% accurate, but it's a good approximation as to what trivial and non-trivial mean.)
Re:um... (Score:5, Funny)
(/me lobbies for the changing of "Off-Topic" moderation to "Orthogonal to Topic")
Re:um... (Score:4, Insightful)
(http://slashdot.org/~Asprin | Last Journal: Wednesday November 05 2003, @03:24PM)
Please, I get enough of that kind of attitude on TV,
in movies,
at work,
at home,
at church,
at the bus stop,
in stores,
at the DMV,
in restaurants,
under my bed
and standing next to ANYONE who has vowels in their names.
Thank you.
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.
Re:um... (Score:5, Insightful)
So if it could be used to break encryption keys as you say, we would not need a proof to start doing so. We could just use it now to generate keys; and its effectiveness would be evident. So no, I don't think that the proof of Riemann's Hypothesis has any sort of bearing on encryption algorithms.
Now it could be that some of the techniques used in the proof itself could provide some insights into prime factorization methods. But again, we don't need a proof itself to get those insights, we just need the techniques themselves.
Re:Okay, assuming this proof to be correct... (Score:5, Insightful)
Not everything need have immediate application.
Of course (Score:5, Funny)
It also proves that all non-trivial zeros are in the line Re(s) = 1/2. This is important because it humbles people without a very wierd Mathematical background, by informing them thre is such a this as trivial and non-trivial zeros. It may also get the Math guys some more girls.
Re:Okay, assuming this proof to be correct... (Score:5, Interesting)
(http://arvindn.livejournal.com/ | Last Journal: Monday June 16 2003, @12:39AM)
It is undeniable that a good deal of elementary mathematics-- and I use the word 'elementary' in the sense in which professional mathematicians use it, in which it includes, for example, a fair working knowledge of the differential and integral calculus) has considerable practical utility. These parts of mathematics are, on the whole, rather dull; they are the parts which have the least aesthetic value. The 'real' mathematics of the 'real' mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly 'useless'(and this is as true of 'applied' as of 'pure' mathematics. It is not possible to justify the life of any genuine professional mathematician on the ground of the 'utility' of his work.
Hardy says that pure mathematics is completely useless. The sweet irony is this: Hardy was a number theorist. In his time, no one could ever conceive that there would ever be any application of that field of mathematics. However, public key cryptography, which was born in 1976, is built on number theory, and is the foundation of modern information privacy and computer security. Immensely practical.
See how it works?
So no, no practical applications for you, but this would still (if correct) be a result of enormous impact.
Re:Okay, assuming this proof to be correct... (Score:5, Informative)
(http://www.fas.harvard.edu/~cudney)
Re:Okay, assuming this proof to be correct... (Score:5, Interesting)
(Last Journal: Saturday September 04 2004, @10:35AM)
The coolest thing that ever happened to me in University (not involving social life), was when we started to prove things that I just took for granted as true.
Suddenly an order and majesty came out of all of it, and it was the more invigorating feeling I've had. There's something to be said about being good at math and able to memorize all of those formulae and how they work, etc. But there is something completely different about proving those formulae and knowing for a fact (beyond any doubt) that they are absolutely true.
Everyone generally assumed RH was true, this is exciting because if it is valid (I don't have the time to validate the proof, albeit I will read it over), than RH is absolutely true beyond any shadow of a doubt.
Now if RH were proven to not be true, that would be even more exciting, but this is just as good. ^_^
Re:Okay, assuming this proof to be correct... (Score:5, Interesting)
(Last Journal: Monday January 06 2003, @10:36PM)
This is not a "most fundamental theor[y]" on which calculus is based. Calculus is not based on it at all. Ostensibly it has nothing to do with calculus at all although any proof will almost certainly use calculus.
You're also confused about the words "theory" and "theorem". We're talking about the latter here. A theorem is a proposition that has been rigorously proved by deriving it from axioms. A theory is something quite different: loosely is means something like a "systematic body of knowledge". Like the theory of evolution or group theory. Or it can be used to mean a tentative hypothesis as in "I have a theory that this doesn't work because you forgot to ...". (That's two distinct meanings by the way - I might as well clear up some Creationist FUD while I'm at it.)
And what are you talking about when you say "proofs are rarely meant to be practical". The truth or falsity of Rimemann's Hypothesis affects things like the theoretical expected time for things like factoring algorithms to run. Maybe you can't see the consequences of that but I'm sure most /. readers can.
GREAT SCOTT!!! (Score:5, Funny)
Wow ... (Score:3, Informative)
-- shayborg
No one noticed this? (Score:5, Interesting)
(http://www.fas.harvard.edu/~cudney)
The arXiv will post nearly anything that resembles a mathematical paper-they don't do any refereeing. However, they apparently use the "general mathematics" section for papers that seem crankish like this one. And the fact that it took more than six months for this proof to make the news is proof that absolutely no one reads that section.
I haven't looked at the proof yet, but I'm worried that it will be at best a "physicist's proof"-a series of claims deduced by using some sort of physical reasoning that is not mathematically rigorous, since it seems to have been written by physicists, and is in the physics section.
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.
Translation (Score:5, Funny)
(http://slashdot.org/ | Last Journal: Wednesday April 16 2003, @07:07AM)
Classic matteproblem able have got a solution
One of mathematics superb olösta problem, the so call Riemannhypotesen, able now have got a solution, 144 year after that that Bernhard Riemann publish sina mind if a special equation, as in its luck is förknippad with primtalen.
If the really am exposing themselves that certificates am holding able they two problemlösarna so småningom give a receipt out one million dollar in reward.
Year 1900 was gathering the world most outstanding mathematics to a conference in Paris. Wonder that present David Hilbert, the terms rankings mathematical think, 23 problem as sedan arrived that affect mathematics wonder heal 1900- digits, and than today. One of these problem each Riemannhypotesen, and defiance superb efforts has it stay olöst. IN November 2002 publish yet Carlos Castro from Clark Atlanta University, Atlanta, America, and Jorge Mahecha from University perceive Antioquia, Centre queue, Colombia a one proposal to solution.
The reward on one million dollar as exhibitor of American Clay Mathematics Institute sheep they yet nots out traitor one year behind publication. This for that second mathematics bark poll term that police accomplishment and watch if the really is accurate.
- About this is truly is the a grand sensation. This is a creature of problem as am claiming great effort entrance a eventual solution able verify. Tusentals mathematics the world over will pounce this and inspect certificates with a magnifier, says Andes Karlqvist, mathematics, professor in data and manager for Polarforskningssekretariatet.
He am declaring that certain of Hilberts problem rather is problemområden than separate problem. A bit had also word if under these term as gone. If now Riemannhypotesen is absolved so is tens of they 23 problems absolved, seven is olösta, five is part absolved and one is nots inferior current.
Under Andes Karlqvist each Hilbert really grand within sits precinct, with him was concluding a epok. He each the lastly as had survey over heal the mathematical science.
Mathematics have the latest decade deployed very quickly, and the cheers article one aid as it olds 1900- digits mathematics nots be able anticipate datorn. Day all prompt and major datorer able manipulate huge amount speech and on short term make computations as formerly each impossible for a mans although he/ she was working a good deal currency with sina figure.
With datorernas help had certain problem absolved, as fyrfärgsproblemet. The says that the nots ring up to more than four various colours for that färglägga a maps so that nots area with a common limit had same colour. One datorprogram had systematic gone through all conceivable alternative.
Andes Karlqvist deem yet that the find one philosophy dilemma with this: inquiring is if husband bark accept evidence in form of one datorprogram. He am believing that wes now am standing before one kulturskifte within mathematics. Wonder the next decade am arriving the that evolve radically, and the because they all efficient datorerna.
Bengal Jonsson
From a Swede (Score:4, Informative)
Slashdot them to hell. It's my university, they can take it.
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:Translation (Score:5, Funny)
I suppose if I were about to win a million dollars, my luck would be förknippad with primtalen, too.
You can help (Score:5, Interesting)
(Last Journal: Friday June 08 2007, @01:42PM)
The proof is... (Score:5, Funny)
Don't get too excited yet... (Score:5, Informative)
(http://www.acm.jhu.edu/~kip/)
Not first millenium prize? (Score:3, Interesting)
(http://www.coralbark.net/)
Statement of the hypothesis (Score:5, Informative)
(http://arvindn.livejournal.com/ | Last Journal: Monday June 16 2003, @12:39AM)
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.
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 o