Riemann Hypothesis Proved?

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.

Riemann hypothesis

First Proof?

um...

"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...

This [dgl.com] provides a pretty good layperson's overview of the problem, as well as why it's relevant.

-- shayborg

gosh, gee willakers

This [dgl.com] provides a pretty good layperson's overview of the problem, as well as why it's relevant.

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!

No, silly rabbit! They don't mean a straight line in the {z(s), s} coordinate plane, they mean a straight line in the {s.r, s.i} coordinate plane.

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.

A proof of the Riemann's hypothesis (RH) about the non-trivial zeros of the Riemann zeta-function is presented.

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.

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.

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...

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...

Construction Boss: cross product you moron! The dot product of zero between the nail and the board means you don't have a board, or you don't have a nail in which case I'll have to spring for workman's comp...but this is canada...so go for it!

Re:Attempt at putting it in more layman's terms.

non-trivial zeros is like the following.

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...

Your comment is entirely orthogonal to the main story, please remain on topic.

(/me lobbies for the changing of "Off-Topic" moderation to "Orthogonal to Topic")

Re:um...

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.

/. is, in fact, the only place since grad school where this sort of news topic discussion is **possible**, let alone normal, so I would appreciate it if you would NOT remind me how much more life is like Empty-V than The Discovery Channel, mmmkay?

Thank you.

Re:um...

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...

if we could rationalise prime numbers, PGP encryption would suddenly become (overnight) about as strong as a password on a zip file, or a world readable /etc/shadow. you could calculate anyones private key from their public one.

I don't buy this at all. Riemann's Hypothesis has been been thought to be true, or at least very likely, for a long time, because it has proven true by example. For example, we know that it's true for the first 1.5 million primes; we know that it's true for at least 40% of all primes.

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...

16-year-old John Quincy Adams and 77-year-old Ben Franklin had watched as Alexandre Charles tested an unmanned hydrogen balloon in Paris. That was where someone asked Franklin what use this all could be, and he gave his much-quoted answer, "What good is a new-born baby?"

Not everything need have immediate application.

Of course

First, it proves Riemann was right.

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...

No, pure mathematics doesn't work like that. Almost no pure math research is driven with practical application in mind. However, what is discovered today may find application 50 or 100 years or even centuries later. As an example, consider Hardy's quote in the "Mathematician's Apology":

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...

The Riemann hypothesis has nothing to do with the foundations of calculus, or what most undergrads learn. Maybe he's thinking of the continuum hypothesis(which was proven to be undecidable). The Riemann hypothesis is a conjecture in analytic number theory. What it gives us is the best possible information on the distribution of prime numbers among all numbers. One common heuristic interpretation of the Riemann hypothesis is that the primes are as randomly distributed as possible. How is this useful in the real world? For instance, the proof that primality could be tested in polynomial time that came out last summer made use of difficult analytic estimates to show that the algorithm would give an answer quickly enough. Agrawal was able to avoid using the Riemann hypothesis, but presumably the Riemann hypothesis could allow one to get a better bound on the time the algorithm takes to run. That's just one example.. the Riemann hypothesis would allow the solution of many other more easily statable conjectures in number theory. For instance one can use it to prove the odd Goldbach conjecture-every odd number greater than five is the sum of three primes.

Re:Okay, assuming this proof to be correct...

That's why I am a mathematician.

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...

I'm sorry but you really don't know what you are talking about and should have been modded down, not up.

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!!!

Finally, the mathematics we need to generate the 1.21 jigawatts to go ...

Wow ...

IIRC this has been something of a holy grail for mathematicians for years. When John Nash tried to prove it in his prime, everyone laughed at him, not believing that even someone as brilliant as Nash could solve the infamous problem. If this isn't a fake, it's a pretty amazing achievement.

-- shayborg

No one noticed this?

This was posted in August, and no one noticed until now? I check the arXiv every single for new articles. But I never saw this. The reason: it was posted in general mathematics and high energy physics theory, not the number theory section where it would belong.
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?

Your worries are well founded. Mathematicians have seen it, and have dismissed it as being very sloppily, and _non-mathematically_ written. Noone I know (mathematicians, that is) have bothered to read beyond the point where they throw their hands up in the air and cry "this is nothing but sloppy handwaving".

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

Here's a "translation" from InterTran [tranexp.com]. Be gentle.

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

Here's a slightly better one [nada.kth.se].

Slashdot them to hell. It's my university, they can take it.

Human Translation

That computer translation is hilarious!

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

its luck is förknippad with primtalen.

I suppose if I were about to win a million dollars, my luck would be förknippad with primtalen, too.

You can help

In case you're tired of looking for UFOs with SETI, you can use your spare CPU cycles to help prove/disprove this hypothesis here [zetagrid.net].

The proof is...

...trivial and left to the reader. Sorry ;-).

Don't get too excited yet...

This item is a submission to a mathematics journal, it has not yet been accepted for publication. The authors claim to have proven the RH by a new method, and the editors of the journal are currently reviewing the article for errors. If they find no major errors, and the minor errors are fixed, then the article will appear in the journal. Then a much larger group of mathematicians will review the article. If, after all this time, the article hasn't been disproved, then it can be accepted as valid.

Not first millenium prize?

I think that the proof of the Poincare Conjecture [arxiv.org]. Will get the first Millenium Prize (by a few days!) if it is correct (talked about here [slashdot.org] but hey if this gets the second then blimey! Two already!?! Could be a good decade for mathematics!

Statement of the hypothesis

Definition of the zeta function: There is something called Riemann's zeta function: it is a function of a single complex variable. It is defined as zeta(z) = 1^(-z)+2^(-z)+3^(-z)+... (ad infinitum) You can easily see that zeta(z), as defined, converges if and only if Re(z)>1 (real part of z). However, the function is defined for all complex z using something called the analytic continuation: basically there is a unique way to extend zeta(z) for re(z) < 1 in such a way that derivatives of all orders exist at all points.

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

Re:Could you get a bit more arrogant please? (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, ... on the horizontal axis.

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 :)

Heh

I guess John Nash can rest easy, now!

The Riemann hypothesis isn't exactly the most practical of problems, but many people have spent decades working on it (and some have gone insane). It's good that it is finally put to rest.

This proof has already gone down in flames

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):

If you are a university mathematics lecturer who teaches analytic number theory, you might want to consider setting your students the task of deconstructing the more serious of these. They may otherwise never be given any serious attention, which would be a shame.

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.

Re:This proof has already gone down in flames

Yeah, I found both of those link about half an hour after I submitted the article. Still, I'd like to add some corrections.

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:This proof has already gone down in flames

1. SvD isn't an "obscure" Swedish newspaper. It's the biggest, counting readers in if not millions so at least hundreds of thousands.

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. :-)

2. That the proof hasn't been verified yet doesn't mean it can't be correct.

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.

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.

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.

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.

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:

1. Write an incomprehensible paper on the Riemann Hypothesis and sling it up onto the xarchiv.
2. ???
3. Profit from the Clay Math Prize!

Re:This proof has already gone down in flames

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...)

I don't understand what you mean by this, or if you are even being serious. If a newspaper is respected, generally trustworthy and read by a large amount of people on a daily basis, where is happens to be located should of course be of no relevance.

If you mean to say that the reason that the rest of the media isn't catching on is that the only story so far has been in a Swedish newspaper, I don't object. But if you're saying that being run in a non-US newspaper makes the story less likely to be true, I think you are a little confused.

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.

Recently, as in the last couple of years, the most promising contributions to RH related stuff has come from high energy physics, and many people, both in math and in physics, believe that this is the approach that will eventually bear fruit. If indeed a final proof would be more physics than number theory (perhaps the proof of the RH would only be a corrollary of some completely un-number theoretic line of reasoning) wouldn't it be reasonable to publish it in a physics journal? That it has implications for number theory isn't enough of a reason to publish it in a number theoretical journal, mostly because the readers of it would not be able to determine whether it was correct or not.

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.

This is serious, agreed. I don't think I read that post.

Unfortunately, the Clay Math Prize has kind of made proving the Riemann Hypothesis a "make-money-fast" scheme.

Only for amateurs. But those have been trying to prove it, as well as Fermat's last theorem, Goldbach's conjecture, the twin prime conjecture etc. for a long time already. Serious researchers, on the other hand, very rarely put their reputation at stake if they don't believe they have something of real value. They know flaws will be detected, and they know that they would never win any prize with an unsound proof.

But, like I said (in the write up, even), there is a clear possibility that this isn't the real thing. I only think you're overstating your case.

Re:This proof has already gone down in flames

I used to be in a number theory mailing list, and every week or so the list would get a mail from some amateur who thought they had proved this or that Longstanding Conjecture (TM). Everybody's favorite was the Goldbach conjecture, with the twin prime conjecture coming in a close second. Usually anyone who with a single number theory course in college could spot the error in a 10-20 line "proof". Sometimes it would be somewhat longer MS-word document that weren't so obvious. Proofs written in TeX were rare, and would need a number theorist to point out the flaw.

There must be hunderds of these "final proof of Riemann hypothesis" claims on the web. It is sad that "a Swedish newspaper is the only one to take up the story yet" doesn't inspire caution in the /. editors but urges them on to more recklessness.

Re:your message is only flamebait

It looks as if you haven't read the thread you pointed us to.

Now, I really did. My favorite quote from it is the part where Aaron Bergman notes, "I also hope that math people realize that us physicists only read Castro's papers for humor purposes."

The mathematicians (and abashed physicists) seem surprised by various techniques and idioms in the proof, but after some good-faith research, they find supporting documentation which they accept.

That's not what I see. What I see is that they figure out among themselves that some really non-standard usages of mathematical terminology happen in physics, and that whether you use "ln" or "log" to refer to the notion of a natural log might reflect where you went to school or what calculator you used...idle chit-chat, really.

And the "clincher" that you cite is a plea for manpower to analyze some possible proofs that will otherwise go unanalyzed, as the rest of the math community is also slavish to the idea of "if it's true, why hasn't it already been proven".

No, my "clincher" would be that a link to this appears on the same page as work by the illlustrious Archimedes Plutonium. [newphys.se] Really, you just don't know how damning this is, do you?

An Explaination of what this means

Can be found here [claymath.org]

What is it good for?

Although the way it's formulated is fairly complex, it is intended to answer the question:

How many prime numbers are there less than a given number.

It doesn't take much thought to work out why that would be handy in say cryptography.

But most complex maths starts for it's own sake. You build the tools in the knowledge that eventually someone is going to use them, and inevitably they always do.

I read about advances in nano-technology all the time. What's the point, no-one's using them? But without them now we wouldn't have cool stuff in 20 years. Same goes for maths. I would have thought nerds of all people would get that point.

oh Riemann you're so fine...

"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."

That is the WORST pickup line I have ever heard.

*siiiigh*

It seems for every bit of knowledge I gain, I just learn how much more there is to know.

Maybe in yet another two or three years I will be able to understand WTF they are talking about. . . .

But....

will this help me get higher frame rates in Doom III? If not then it is useless ;)

Eu4ria

So if this is proven to be true...

Will it lead the way to the return of an integer to represent your karma?

Re:Crypto applications

(1.5*10^9 examples aren't often wrong)

read the sales figures for pop albums.

Watertight Proof?

so there is certainly a possibility that this is a hoax, or a less than watertight proof.

I read this as waterproof tights for some reason... had visions of Batman in a wetsuit.

Explained With Pictures

The theorem is explained here [ou.edu] with pictures.

Mathworld comment about the prize

from Mathworld [wolfram.com] and I quote:

In 2000, Clay Mathematics Institute offered a $1 million prize for proof of the Riemann hypothesis. Interestingly, disproof of the Riemann hypothesis (e.g., by using a computer to actually find a zero off the critical line), does not earn the $1 million award.

An example that operates on the exact opposite principle of awarding prizes is the recent battle between Kasprov and Deep Jr: He gets 500k regardless, and 300k extra if he wins, 200k extra if he loses, or 250k if draw (i think the last case took place).

Talk about being stingy! I'd think that disproving the Riemann Hypothesis would be equally interesting as proving it - There are soooo many theorms out there that basically begins with "We assume that the Riemann Hypothesis to be true, and so forth so forth."