Mathematician Claims Proof of Riemann Hypothesis

TheSync points to this press release about a Purdue University mathematician, Louis de Branges de Bourcia, who claims to have "proven the Riemann hypothesis, considered to be the greatest unsolved problem in mathematics. It states that all non-trivial zeros of the zeta function lie on the line 1/2 + it as t ranges over the real numbers. You can read his proof here. The Clay Mathematics Institute offers a $1 million prize to the first prover."

If there's one thing I know

It's that mathematicians love to exaggerate! Like infinity is infinite, or pi goes on forever! Those guys are always talking big.

homer simpson

mmmmmmmm......infinite pie..!

Re:If there's one thing I know

Next he'll be solving problems that are NP-Complete. We'll have to re-write all our textbooks!

Not to spoil your joke or anything, but actually, AFAIK, NP-complete problems are perfectly solvable. The problem is how long it takes to solve them in general (a certain instance of a problem could prove easy). They cannot be solved deterministically in polynomial time (i.e., quickly).

Re:If there's one thing I know

The definition of NP-Complete has nothing to do with whether the problem can be solved in polynomial time.

Not quite correct, because:

The set NP is defined as all problems whose solutions can be verified in polynomial time

...is one of *two* equivalent definitions of the class NP. The other is:

"Set of all problems which can be solved in polynomial time by a nondeterministic Turing Machine."

So the question "Does P equal NP?" is also the question "Does an NTM have the same computional power(in time) as a TM, or does it have more?" (It is already known that as far as decidiblity is concerned, TMs and NTM are equivalent.)

Because of the way the proofs have been constructed, if you can solve any of the NP-Complete problems in polynomial time, you can solve all of them in polynomial time.

Some more detail here:

An NP-hard problem is a problem to which any problem in NP can be reduced in polynomial time.

(Essentially, it can be used as a subroutine for any NP problem, with only a polynomial number of calls. Thus a solution to it is a solution to any problem in NP.)

An NP-complete problem is one that is:

1. NP-hard
2. In the set NP.

Thus if a polynomial-time solution exists to an NP-complete problem, then P=NP, because a polynomial number of calls to a function that terminates in polynomial time is O({polynomial}*{polynomial}) = O({polynomial}) .

Please note, however, that not all NP-hard problems are NP-complete.

Re:If there's one thing I know

NP-Complete problems are by definition problems that can't be solved in polynomial time(at least not by a Turing machine?). However, most problems that are considered NP-Complete are not mathematically proven to be so. Some are though, and the thing with NP-Complete problems is that you can always translate one NP-Complete problem to another NP-Complete problem.

Sorry, but you got things mixed up.

NP problems are problems that can be solved by a Nondeterministic Turing machine in Polynomial time. NP-Complete problems are the class of "hardest" problems in NP. All the usual suspects (Traveling Salesman, 3-SAT, SAT, ...) are proven to be in NPC.

We know that we can solve NPC problems in exponential time (as we can simulate a non-deterministic Turing maschine on deterministic hardware with exponential overhead). What we do not know is if there is any smarter way. That is the P=?=NP question.

Re:If there's one thing I know

NP-Complete problems are by definition problems that can't be solved in polynomial time(at least not by a Turing machine?).

No, you're wrong. NP problems are, by definition, problems that can be solved in polynomial time by a nondeterministic Turing machine [wikipedia.org].

Essentially this means that a Turing machine could solve the problem in polynomial time, if it had some magic 'oracle' which instructed it on the right computational path to follow for a given input.

Obviously there are problems out there that would require exponential time for even a nondeterministic Turing machine to solve. An example from the Wikipedia link I provided is finding the best move in a chess or Go game.

Such problems are not in NP, and proving P=NP would not suddenly give us algorithms for solving these problems deterministically in polynomial time.

However, most problems that are considered NP-Complete are not mathematically proven to be so.

What? Sorry, if there ain't a proof, it ain't NP-complete. There are a lot of problems that are described as "believed to be NP-complete", but that's different.

Re:If there's one thing I know

De branges is a bit of a crank on the Riemann hypothesis. No-one believes his approach(s) will work. This is well documented in the book "Riemann's Zeros". When some of the leading mathematians were asked about his approach they said it was "full of errors" and "unlikely to work". The only reason he is given the light of day is because he managed to prove to the Bieberbach conjecture. That was a difficult problem, hats off to him for getting it aswell, but it's no Riemann hypothesis!

Rest assured, we'll all be dead and burried when it actually gets solved.

Simon

Re:If there's one thing I know

It would appear that mathworld.com agrees with you...

----------------

Riemann Hypothesis "Proof" Much Ado About Noithing
A June 8 Purdue University news release reports a proof of the Riemann Hypothesis by L. de Branges. However, both the 23-page preprint cited in the release (which is actually from 2003) and a longer preprint from 2004 on de Branges's home page seem to lack an actual proof. Furthermore, a counterexample to de Branges's approach due to Conrey and Li has been known since 1998. The media coverage therefore appears to be much ado about nothing.

Apology

Apology for the proof of the Riemann hypothesis (in pdf format). [purdue.edu]

"We humbly apologize for the complete illegibility of this proof. The mathematician responsible has been sacked."

Re:Apology

> "We humbly apologize for the complete illegibility of this proof. The mathematician responsible has been sacked."

"A Slashdotter has discovered a truly wonderful proof of the sacking of the mathematician responsible, but his bandwidth is too narrow to host it!"

WTF? Mods?

From reference.dictionary.com:

Apology - 2: a formal written defense of something you believe in strongly

This should at most have earned a "Funny", or is there something I'm missing here?

Re:WTF? Mods?

This should at most have earned a "Funny", or is there something I'm missing here?

Yeah, I think you missed:
Equivocation - \E*quiv`o*ca"tion\, n. The use of expressions susceptible of a double signification, with a purpose to mislead boneheaded moderators, especially when you are just making a joke.

Re:Apology

Uh, the above comment was a joke people. The quote in the parent post does NOT appear in the document. Apology in this case means a defense of the proof.

Re:Apology

Note to mods: Mod parent funny, not interesting! This is a play off a quote from the beginning credits sequence in Monty Python and the Holy Grail. As for the pdf link, it's the first link in the purdue page referenced in the article. RTFA, people!

Re:Apology

Of course if I were to RTFA - and more importantly UTFA (Understand the Article) I wouldn't be able to post this for another 2 years or so...

As it is, it looks like he proposed this solution over a year ago and has been getting it vetted in a tightly controlled community. Now that the cat is out of the bag he will have to get it into a peer reviewed journal (takes 6 months or so) and wait 2 years to see how it is bashed...

Yeah - that is about the time it would take for me to UTFA, except I am not a Mathemetician, so add in another 6-8 years to get that training as well. So I will get back to you sometime around 2120 with an insightful comment after UTFA

Re:Apology

The last paragraph of the article is interesting:

A curious coincidence needs to be mentioned as part of the chain of events which con-

cluded in the proof of the Riemann hypothesis. The feudal family de Branges originates in

a crusader who died in 1199 leaving an emblem of three swords hanging over three coins,

surmounted by the traditional crown designating a count, and inscribed with the motto

"Nec vi nec numero." This is a citation from Chapter 4, Verse 6, of the Book of Zechariah:

"Not by might, nor by power, but by my Spirit, says the Lord of Hosts." The chateau de

Branges was destroyed in 1478 by the army of Louix XI of France during an unsuccessful

campaign to wrest Franche-Comte from the heirs of Charles the Bold of Burgundy. The

family de Branges performed administrative, legal, and religious functions in Saint-Amour

for the marquisat d'Andelot during Spanish rule of Franche-Comte. Francois de Branges

of Saint-Amour received the seigneurie de Bourcia in 1679 when Franche-Comte became

part of France. The chateau de Bourcia remained the home of his descendants until it was

destroyed by Parisian revolutionaries in 1791. The chateau d'Andelot near Saint-Amour,

which survived the revolution, was bought in 1926 by Pierre du Pont, an elder brother

of Irenee du Pont, for a nephew assigned in diplomatic service to France. This coinci-

dence accounts for the interest which Irenee du Pont showed in a student of mathematics.

The ruin of the chateau de Bourcia overlooks a fertile valley surrounded by wooded hills.

The site is ideal for a mathematical research institute. The restoration of the chateau for

that purpose would be an appropriate use of the million dollars offered for a proof of the

Riemann hypothesis.

That's quite noble of him.

Re:Apology

The Bourcia Mathematical Research Institute will involve more whores and cocaine than a typical research institute, but for tax purposes it's a research institute.

-B

Re:Apology

This guy is an all around class act. I've always found mathematicians to be kind of standoffish, and while this guy is obviously at the top of his field, he's also on top of the rhetorical game, the very structure of this "Apology" shows that he's having a great deal of fun with his chosen profession.

My favorite selection:

The solution of a celebrated problem creates a disturbance in the otherwise quiet flow of mathematical events. The solution escapes the planning of committees. Colleagues are unprepared because the possibility of a solution has not been included in their research proposals. Students have avoided related thesis topics because of the risk that the work will not be welcome to a prospective employer. Friends are discouraged from research activity by the demands of the situation created by the solution. The manuscript, which is necessarily written at the highest research level, is readable only to a limited audience. An introduction is therefore needed which makes available the opportunities created by the solution. This is done by supplying motivation for the argument in a chronological order which also gives an account of how the solution was obtained.

Hilarious stuff. He apologizes to the people who will now feel the need to go over his proof with a fine toother comb, looking for mistakes...and also explains (three pages in) why he's chosen to start his proof with a history of the golden age of mathematics, stretching back to Newton. Basically, he's saying "oh hey, thanks for joining me. I was just explaining ALL OF MATHEMATICS for those playing at home. Bear with me, this one's worth it, and I promise you can get back to your euclidian algorithms and Ving diagrams in short time."

Ever read "The Life and Opinions of Tristram Shandy?" It's an amazing book from the 18th century, which attempts to tell a simple narrative but due to the extremely schizophrenic style of the narrator, it keeps breaking down into tangential pockets of narrative self awareness. Basically, the author wrote from the perception of a disturbed dandy who couldn't keep his mind on the task at hand, an author who keeps apologizing to his readers for the inconvenience of his own poor editing.

This mathematical proof reminds me a lot of this book...the text of the proof doesn't act as though the proof isn't something interesting or ground breaking, nor does it make a big deal of this. It just ambles on in all directions until the Riemann hypothesis is well and truly proven, but with no real hurry to illustrate the proof until the outlines have been inked. Not that I know for sure that Riemann is proven or isn't...my brain was full when I got to differentials. But if it is, this paper will stand out not only as a great work of mathematics, but a great work of WRITING about mathematics.

I'm going to read it again. Maybe I'll understand it this time!

Well, not exactly

Sorry, but...

It is not proved; he is not at the top of his field; this "paper" will be quickly forgotten among professional mathematicians; and I doubt any professional mathematician is going over the proof with any sort of comb.

L. de Branges first achieved fame for proving the Bieberbach conjecture [wikipedia.org]. His proof went through strange and abstract methods. He went on the road to present his proof at various seminars in France, Russia, etc; IIRC a bunch of Russian students got very excited and basically rewrote his proof. Their new proof was much shorter and avoided the use of strange methods. Nowadays, their proof is remembered and his is not, but the proof still bears his name, since after all he was the first to come up with *some* kind of proof, and their proof did more or less come out of his.

So he deserves credit for that, and it was quite an achievement to prove the Bieberbach conjecture. But even then he was using unwieldy proofs with unnecessarily abstract methods.

For many years he has been claiming to have a proof of the Riemann Hypothesis. Professional mathematicians stopped listening a long time ago.

This guy is washed-up.

I whole-heartedly agree that this short article is hilarious, but I would like to add the adjective condescending. What kind of asshole apologizes for solving a problem? Does he think he lives on some higher plane, and therefore must take direct, personal responsibility for every aspect of our lives?

Look at how G. Perelman submitted his ideas on proving the Poincare conjecture [wikipedia.org] just a little while ago. He didn't waste anyone's time by rehashing the already-available history of the problem or its wider context in mathematics. Nor did he apologize for having an idea. Rather, he submitted his ideas for consideration, with the full awareness that there may have been a mistake. .... Now, this is where I admit that I do not really understand that area of math, and have not been closely following the status of (alliteration alert) Perelman's proposed proof. Still, Perelman is a real mathematician, and even if the proof is (was?) wrong, it has real ideas of value in it.

de Branges is so full of crap, it makes me sick.

zach

Good job

It's too bad that most of society does not recognize truly great achievements like this. I, for one, admit interest but not enough knowledge of the details to read and understand the proof. I'm sure most people here on /., as representatives of the intelligent future of sentient life, have the interest as well.

Re:Good job

You're probably right. But society does recognize a one million dollar prize. This one may actually get TV time. Funny how that works.

Gotta prove 'em all

They really should make mathematics more like pokemon, it would get more people interested in the subject
Riemann-chu, I prove you! Then bust out the paper.

Re:Gotta prove 'em all

Mathomon?
Yeah but then a few years later Yu-Physics-Oh comes along and replaces it in popularity. Then before you know it that two is gone replaced by annother populare science.
Plus it would replace Arceology the gathering.

Magic The Gathering, Pokemon and yugioh are in the 15 minuts of fame catagory. Populare today gone tomarow.

I don't want Math to be gone tomarow. I'm counting on it to stay for a while.

Now english I wouldn't mind if it's own end was spelled out. You can see the proof reading this very post.

Nope! Nice try

I read through his proof and...nope, it's wrong. I know the real answer, but am leaving it as an exercise for the interested student.

For some suggested approaches, see

http://www.maths.uwa.edu.au/~berwin/humour/invalid .proofs.html [uwa.edu.au]

Failed proof

Ha! They've already found an error in the proof! All that he posted was his apology! [purdue.edu] :-)

Yes, I was actually confused at first. For the non-math geeks like myself, who are feeling stupid, look at definition 2a of apology [reference.com].

Uh-oh! There's a mistake!

I don't want to give it away, but you'll see it.

Hilbert Turns in his Grave?

"If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?"

David Hilbert

Re:Hilbert Turns in his Grave?

Hilbert may have been referring to the importance of the Riemann Conjecture, and not the difficulty of proving it.

Really, folks, this is a big deal if it's true. It just doesn't get the attention Fermat's Last Theorem did because it's harder to understand what it means and why it's important.

After all, most people don't even know what complex numbers are, much less complex functions. The zeta function, then, is already beyond the understanding of most people, not because they're incapable, but because they're not interested. But the implications of the Riemann Conjecture are far-reaching indeed, affecting things like quantum mechanics and statistical physics.

--Mark

Re:Hilbert Turns in his Grave?

"If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?"

Oh yeah? Mine would be "Is Doom 3 out yet?"

Honestly, which is more likely?

I'm in trouble

You know you're in trouble when you don't even understand the question.

The question, explained

First: complex numbers [wolfram.com], explained. You may have heard the question asked, "what is the square root of minus one?" Well, maths has an answer and we call it i [wolfram.com]. i*i = -1. If the real number line ...-4, -3, -2, -1, 0, 1, 2, 3, 4... is represented as a horizontal line, then the numbers ...-4i, -3i, -2i, -i, 0, i, 2i, 3i, 4i... can be thought of as the *vertical* axis on this diagram. The whole plane taken together is then called the complex plane [wolfram.com]. This is a two-dimensional set of numbers. Every number can be represented in the form a+bi. For real numbers, b=0.

Right. Now the Riemann Zeta Function [wolfram.com] is a function/map (like f(x)=x^2 is a function) on the complex plane. For any number a+bi, zeta(a+bi)will be another complex number, c+di.

Now, a zero [wolfram.com] of a function is (pretty obviously) a point a+bi where f(a+bi)=0. If f(x)=x^2 then the only zero is obviously at 0, where f(0)=0. For the Riemann Zeta Function this is more complicated. It basically has two types of zeros: the "trivial" zeroes, that occur at all negative even integers, that is, -2, -4, -6, -8... and the "nontrivial" zeroes, which are all the OTHER ones.

As far as we know, *all* the nontrivial zeroes occur at 1/2 + bi for some b. No others have been found in a lot of looking... but are they ALL like that? The Riemann Hypothesis [wolfram.com] suggests that they are... but until today nobody has been able to prove it.

Re:The question, explained

It seems like the answer (well, we'll call it "i") has been proposed before anyone has shown if can really happen.

Great Cthulhu help me, but I'm going to try and answer this for you.

We have natural numbers - 1,2,3, ... - and people are happy with this. It's an abstract way of representing a real property. I have five oranges, I owe you four oranges. Natural.

And then we have Zero and once upon a time this disturbed people. You grew up with it, you're happy with it; but we can see that it was less intuitive than 1,2,3, ... because it developed so much later and the greeks managed without it for quite a long time. It's not an abstraction in the same way that these other numbers are. People used to ask questions such as, 'how can something exist and yet be nothing?' 'How can zero x zero = zero since that means you have no zero's?' Can you prove that it does mathematically, right now? *

And yet, the discovery (or creation ;) of Zero allowed people to abstract in new ways that produced real world results. The same can be said of Negative numbers which are even less intuitive. If I give you those four oranges mentioned earlier (not bloody likely since I'm writing this before breakfast), then that leaves me with one. But suppose I owe you six oranges? We can't carry out that operation with oranges, but the operation is useful in many other areas, the most obvious is probably money. You can be overdrawn for example - that's applied negative numbers. Is there really anti-money in your account? Well, yes, why not? It's just numbers, and numbers are an abstraction, a model of something if you like. It's perfectly normal to represent some properties as negatives. Try basic Newtonian physics - two bodies moving in opposite directions towards each other. You treat the momentum of one of them as negative and the other positive which lets you work out which direction they're going in after collision.

Now perhaps at this point, you're nodding and saying 'yes, yes, I know that already.' If so, then good, because you've just understood the principle of a complex number. It's another abstraction that can't easily be represented in the real world (nuclear physicists shut up, please). And yet, it has very real use in making calculations.

If you're a programmer, think about how much code there is behind the scenes of a program to produce the result you want from it. Suppose that your program counts how many oranges people have given you. Maybe it has the line
for (i=0; i &lt oranges_owed; i++) {}

Well i isn't physically real, it doesn't represent a physical aspect of what you are modelling (the oranges) but it's useful. And in the same way, i is also useful, even if it's just part of a intellectual model.

For a mathematician: I think therefore i is.

The only thing remaining is to give you an example of how it is useful. Easily done - Quantum Physics. All of it. ;)

Hope this helps, IASNAM (I Am Surprisingly Not...)


* Proof that 0x0=0:

0=1x0

0=(0+1)x0
0=0x0+1x0
0=0x0+0
0=0+0x0
0=0x0

Oh Hocky Sticks!!!!

There's the occasional post that deserves to be modded to "+10 -- Best Damn Thing I've Read On Slashdot This Year".

Thanks!

Is it...

... 42?

Impact on crypto?

This theorem is a theory of how prime numbers are distributed...so does it's proof have any impact on crypto? Does it make it any easier to find prime numbers?

There is no impact on crypto

The Riemann Hypothesis, among other things, implies that the Prime Number Theorem is off in the distribution of primes by no more than O(sqrt(n)*log(n)). However even without the full result, we already had very good error bounds for the approximation of the prime number theorem for "small" numbers, including numbers far larger than any which come up in cryptography.

Re:Impact on crypto?

The art of cryptography can be summed up as: Easy to encode, but hard to decode.

Prime numbers are easy to multiply together. Little CPU needed.

But it's hard to do the reverse: Factor a big number into two separate prime numbers. Lots of CPU needed.

It's based on that principle.

Re:Impact on crypto?

One of the fallout corollaries from a proof of the Riemann hypothesis is that there exists a simple algorithm for factorization (read: p-time).

No. GRH implies that isprime() is in P (by bounding the cost of a strong pseudoprime test); but we already knew that, thanks to AKS.

Re:Impact on crypto?

This theorem is a theory of how prime numbers are distributed...

It's actually a little more complex than that.

Riemann was investigating the distribution of prime numbers. Along the way he devised (discovered?) the Zeta Function, which describes with considerable accuracy the distribution of prime numbers. While working with the Zeta Function, he discovered an interesting property: It appeared that all the non-trivial zeroes of the function had a real part of one-half. However, since this property of the function was not related to the prime-distribution work he was doing, he did not bother to prove this apparent property, which came to be known as the "Riemann Hypothesis" (presumably, once it is proven it will be known as the Riemann Theorem, or some such).

Thus, the Riemann Hypothesis is in fact tangential to (and possibly unrelated to) the distribution of prime numbers. Riemann's notes on the Zeta Function, regarding his work on prime distribution, are pretty explicit about this.

Re:Impact on crypto?

It's actually a little more complex than that.

*smack!*

Re:Impact on crypto?

Along the way he devised (discovered?) the Zeta Function, which describes with considerable accuracy the distribution of prime numbers.

Actually, as with most things Euler was the first to study it. The zeta function is also the simplest of a class of functions that Dirichlet studied Dirichlet L-series. There is also a Generalized Riemann Hypothesis that states that no Dirichlet L-series has zero with real part greater than 1/2.

The Riemann Hypothesis is more than tangential to the study of the distribution of primes. There is a function derived from the distribution of the primes that can be expressed in terms of the non-trivial zeros of the zeta function. The Prime Number Theorem is also equivalent to the statement that the zeta function has no zeros with real part 1. The Generalized Riemann Hypothesis implies the weak form of Goldbach's conjecture (i.e. that any odd number greater than 7 can be expressed as the sum of three odd primes).

Re:Impact on crypto?

does it's proof have any impact on crypto?

No. Almost all mathematicians have assumed for years that GRH is true anyway; proving it would mean that all those footnotes ([1] Under the assumption of the Riemann Hypothesis) could be removed, but that's the only practical effect it would have.

What are the consequences for cryptography?

Does this affect prime based public key schemes at all? How does it affect them?

Re:What are the consequences for cryptography?

Nope, probably not.
Most mathematicians felt that the Riemann Hypothesis was true so that this view has been taken into consideration into mathematics for a long time. Perhaps if he developed some new methods for playing with numbers in the proof, but it doesn't seem like it to me.
There's a ton of math papers that begin with "Assume the extended riemann hypothesis...".

At least that's my guess.

Apologies to the proof?

I knew it was a hoax when he started discussing his Paley-Wiener space...

The media never learn?

Will the media keep publishing claims of extraordinary mathematical findings without checking the facts forever?

Just like this one over again:
Swedish Student Partly Solves 16th Hilbert Problem [slashdot.org]

That's what I like about /. If the article is wrong, there is always the comments there to solve it.

de Branges' reputation with other mathematicians

Although I hope de Branges has found a proof, I'm not too optimistic. It seems that de Branges has a reputation among mathematicians for going off half-cocked. He does have the Bieberbach proof under his belt, though, so you never know.

Re:de Branges' reputation with other mathematician

He's just trying to disprove the "Field's Medal Hypothesis" -- no one over the age of 40 can accomplish innovative math.

He appears to be 72.

A Proof .... Maybe

It seems that the proof hasn't been reviewed yet. He may have it -- but lots of good folks have tried, without success. This from Science Daily: http://www.math.purdue.edu/~branges/ . While mathematicians ordinarily announce their work at formal conferences or in scientific journals, the spirited competition to prove the hypothesis - which carries a $1 million prize for whomever accomplishes it first - has encouraged de Branges to announce his work as soon as it was completed. "I invite other mathematicians to examine my efforts," said de Branges, who is the Edward C. Elliott Distinguished Professor of Mathematics in Purdue's School of Science. "While I will eventually submit my proof for formal publication, due to the circumstances I felt it necessary to post the work on the Internet immediately."

quick google search

... shows [google.com] that he's been offering "proofs" since July 1989. I see from MathSciNet [ams.org] that he has 87 papers from 1958 to 1994, but isn't this a bit like the boy who cried wolf?

Re:quick google search

Not really. It means he's a prolific member of the community who is not afraid to take risks with his work. Consider an experimental scientist -- in an experiment, one that turns back negative results, or on that fails, still produces important data. Similarly, this is like "experimental mathematics." If he fails, then we'll know why he fails, how far he got doing things right and other things which can point us to the correct proof.

Hm

I think I speak for all non-mathematicians when I say:

what?

Re:Riemann hypothesis proof is useless

Not to take away from the brilliant work of this guy, and I'm sure his work will have generated some good math on the way. But just knowing whether the Riemann hypothesis is true is not of much help (people have been assuming it to be true for a while).

Your comment explains why discovering a proof for the Riemann Hypothesis is such a monumental event. Mathematicians have assumed it to be true for some time now, and there exists a massive amount of mathematical theory which rests upon its validity. Proving the hypothesis ensures that their reasoning is on solid ground. Without one, there's no way to know for sure whether or not their conjectures are true.

Seems not-unlikely to be wrong

Sorry to burst the bubble, but some usenetting shows:

The same guy claimed [google.com] to have solved the same problem at least 4 years ago.
The guy has a reputation [google.com] for sometimes getting it wrong.

(Probably because he has published flawed proofs [google.com] of other well-known problems.)

He could be right, but I wouldn't get my hopes up.

Re:Seems not-unlikely to be wrong

otoh, he proved the Bieberbach conjecture in 84 and has been working on this since. Perhaps this is why he posted it before it is formally published in a journal.

Re:Seems not-unlikely to be wrong

Well, he is reliably credited with solving the Bieberbach conjecture - the guy isn't a complete nut.

However, a quick scan suggests that if his proof is indeed verified, it won't do what a lot of people want it to do: Quote from the article: "The proof of the Riemann hypothesis verifies a positivity condition only for those Dirichlet zeta functions which are associated with nonprincipal real characters. The classical zeta function does not satisfy a positivity condition since the condition is not compatible with the singularity of the function. But a weaker condition is satisfied which has the desired implication for zeros."

So I may be wrong, but it looks like he may have found ground on a restricted interpretation of the GRH (or Generalized Riemann Hypothesis), -ie concerning Dirichlet zeta functions which are associated with nonprincipal real characters only.

As for consequences, If GRH is indeed true, then e.g. the Miller-Rabin primality test is guaranteed to run in polynomial time.

The Problem

The problem is simple enough to understand, assuming you know some math basics. As most of you know, any function f(X) where f(Xo)=0 is said to have a zero at Xo. For functions of complex numbers f(z) where z=x+iy and x,y are real numbers, you obviously have the function taking on different values for every x and y, so the zeros can be anywhere on the x-y plane. For the zeta function, "trivial zeros" occur at the negative even integers (z=-2+i0,-4+i0,...) and also at points on the line x=1/2 (i.e 1/2 +iy for certain y).The Riemann Hypothesis says that all zeros that aren't negative even integers lie on this line.

Most of you have who have taken basic calculus courses have probably seen a simplified definition of the zeta function for real intergers greater than 1. when z=n, a natural number, the zeta function reduces to the infinite series Zeta(n)= SUM (k=1-->inf) 1/k^n

Died before he could prove it

I love this sentence from the article:

The origins of the hypothesis date back to 1859, when mathematician Bernhard Riemann came up with a theory about how prime numbers were distributed, but he died in 1866, before he could conclusively prove it.

As he didn't prove the result, either before or after his death, how can it be said that he died before he proved it? Maybe the lives of great mathematicians form arcs in some abstract space that can be extrapolated beyond their death?

I think I might as well write my epitaph now:

Here lies exp(pi*sqrt(163))

He died before he could get laid by Charlize Theron

Re:Died before he could prove it

"died before he *could* conclusively prove it", not "died before he proved it". The original doesn't have the logical flaw that you've chosen to pick on in your misquoted version.

Re:Died before he could prove it

You're not dead yet. Go for it!

Re:Died before he could prove it

Mathematicans don't die, they just loose some of their functions....

actual paper

The 23 page "apology" is not the actual purported proof, contrary to what the article and press release say. The actual proof is the manuscript "Riemann zeta functions", the third link on de Branges' home page, which weighs in at 124 pages!

So if his "proof" isn't obviously wrong, it'll likely take quite a while for the experts to verify.

Much ado about nothing?

The proof (or, better said, the sketch of the proof) actually starts at the end of page 21, very close to the last page. The original work is actually pretty hard to find since it is buried in so many unrelated side notes.

Here is the general outline:
1) At the end of page 19 he mentions that "The positivity condition which is introduced implies the Riemann hypothesis if it applies to Dirichlet zeta functions."
2) After some introduction of the quantum gamma functions that lasts two pages, the actual proof starts at the end of page 21 with the phrase "A quantum gamma function is obtained when is nonnegative. A proof of positivity is given from properties of the Laplace transformation."
3) The proof ends in the middle of page 23 with the a verification that W(z) is a quantum gamma function with quantum q = exp(-2*pi), obtained from a spectral theory of the shift operator.

Overall this is just a very brief sketch of the whole proof.

BTW, to add gas on fire, here is an exceprt from mathworld.com, which surprisingly was missed by /. until now :-)

http://mathworld.wolfram.com

Riemann Hypothesis "Proof" Much Ado About Noithing (sic)
A June 8 Purdue University news release reports a proof of the Riemann Hypothesis by L. de Branges. However, both the 23-page preprint cited in the release (which is actually from 2003) and a longer preprint from 2004 on de Branges's home page seem to lack an actual proof. Furthermore, a counterexample to de Branges's approach due to Conrey and Li has been known since 1998. The media coverage therefore appears to be much ado about nothing.

The counterexample to Brangles approach can be reached here: http://arxiv.org/abs/math.NT/9812166

Maybe

I looked at de Branges' "Apology for the proof of the Riemann hypothesis" and found no proof. Perhaps the proof is in another document?
Even though he is a kook, I root for him; no one believed him when he claimed he had proven the Bieberbach conjecture. I believe, however, that he has claimed to have proven the Riemann hypothesis previously. One should check carefully before trusting his claim.

It's already been solved...

It's 42.

Besides, I think he forgot to carry the one.

"Apology" an old paper?

The 23-page "Apology" referred to in the press release is also apparently mentioned in this 1996 Usenet post [google.com]. So is there a new proof? No one seems to know yet.

Another Proof

I have another proof Of Riemann Hypothesis but this text area is too small for it, anyway /. comments doesn't allow math symbols.

Cool background material

A cool overview of why this is such an interesting hypothesis [ex.ac.uk].

If nothing else check out the animation [ex.ac.uk].

mind-boggling

Good Book about the Hypothesis

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics [amazon.com]

This is a very informative book about Riemann's work on the hypothesis, as well as the work of many other mathematicians. You probably need a solid college-level understanding of math to follow most of the technical explanations, but the historical parts of the book are very interesting.

Why people haven't believed him so far

As others mentioned, de Branges has been claiming a proof along the same lines for years. He's hard to dismiss because he actually proved the Bieberbach conjecture -- a startling exception in the series of wrong proofs he's been famous for, before and since.

The reasons why most specialists doubt that his approach can ever yield the result are well described in this paper [arxiv.org] from 1998:

In this note, we shall (...) give examples showing that de Branges' positivity conditions, which imply the generalized Riemann hypothesis, are not satisfied by defining functions of reproducing kernel Hilbert spaces associated with the Riemann zeta function zeta(s)

(i.e., despite the name, the "generalized RH" proved by de Branges actually did not include the standard RH as a special case.)

Riemann Hypothesis Interview

Berkeley Groks [groks.net] has an interview [berkeley.edu] that aired today with John Derbyshire [olimu.com] discussing the Riemann Hypothesis. He states that after talking with many mathematicians in the field, the prospects for a solution any time soon are quite low.

Probably a hoax:

mathworld.wolfram.com
Riemann Hypothesis "Proof" Much Ado About Noithing A June 8 Purdue University news release reports a proof of the Riemann Hypothesis by L. de Branges. However, both the 23-page preprint cited in the release (which is actually from 2003) and a longer preprint from 2004 on de Branges's home page seem to lack an actual proof. Furthermore, a counterexample to de Branges's approach due to Conrey and Li has been known since 1998. The media coverage therefore appears to be much ado about nothing

Overview of proof?

Could someone capable in the apropriate math(s) please explain how the proof works?

I found a wonderful proof for this theorem

but unfortunately it was censored by the Slashdot lameness filter.

Re:Proof of theory

Interesting that the only time a proof of concept is ever challanged is when money is involved.

Bull. There are thousands of mathematical researchers. Most don't have hefty salaries, and most aren't working on money-prize problems.

Mathematicians are never in it for the money.

Wonder what he'll do with the money?

Seems like he wants to restore the old family castle:

The ruin of the chateau de Bourcia overlooks a fertile valley surrounded by wooded hills. The site is ideal for a mathematical research institute. The restoration of the ch^ateau for that purpose would be an appropriate use of the million dollars offered for a proof of the Riemann hypothesis.

I must say that at he seems a bit full of himself, or at least, getting a bit ahead of himself. Given how many have tried and failed witht his problem.

Re:Proof of theory

huh?

Mathematicians have been working on this for a long time. it is not like one day this guy woke up and said "oh, 1 million dollars for it eh, well I better get to work."

Re:Already failed once!

So if a guy fails you should never listen to him again?

It took Einstein many tries to arrive at the correct fomulation for general relativity. I guess according to you, he should have just given up after his first failure?

Re:Proof of theory

Dude, what the fuck are you talking about?

Mathematicians tackle difficult problems all of the time, regardless of the (lack of) money involved.

I don't know why you say that interest in "theoretical" mathematical proof is waning. It certainly isn't where I come from. (And what is ultra-math??!)

Re:Proof of theory

Wonder what he'll do with the money?

Purdue will take the money, because he works there. It will be used to build a new scoreboard for the football stadium.

Then the IRS will send de Branges a huge bill for the 45% tax rate on "winnings."

Then his ex-wife will sue for 50% of the million dollars because "he used to moan 'oh, Riemann' while we were doing it."

Then de Branges will spend 25 years opening letters from the poor and destitute who desparately deserve a chunk of his newfound yet nonexistent wealth.

Then eventually he will take his place in an unmarked mass grave reserved for all the great mathematicians who died peniless and unloved.

Well, that's my guess anyways.

re:Already failed once

A long time ago, in the distant past, there were Finders. Dedicated individuals that wandered around outside the camps and found stuff. Over time, it became more difficult to find stuff, and the Finders became the Searchers. Many times the Searchers would return empty handed. As technologies improve and new insights are gained, the same fruitless searches of the past were repeated. Sometimes with a new results, sometimes as fruitless as before. Regardless, it was this not giving up on an idea just because it failed once that led the change in title from Searcher to Researcher.

Most reseachers I know produce one magnificent failure after another on the quest for a new piece of knowledge. Everything that is easy to find has probably already been discovered, and mathematics is no different. So the guy made a few failed attempts at solving the puzzle, this doesn't make each sucessor to the first attempt a garaunteed failure.

Re:Already failed once

A long time ago, in the distant past, there were Finders. Dedicated individuals that wandered around outside the camps and found stuff. Over time, it became more difficult to find stuff, and the Finders became the Searchers.

And so it came to pass, Gentle Reader, that some of the Finders did find their fruit, and these were known as Keepers. But a few still lost their newfound fruit on the way home, and these poor souls were thenceforth known as Losers, unless they wept, in which case they were also known as Weepers.

Re:The Reimann Hypothesis

I think you are going a bit overboard here. The Riemann hypothesis is the greatest open problem in mathematics right now and solving it would be HUGE :-). However, famous open problems usually do not advance mathematics that much and I suspect that a proof of the Riemann hypothesis would not introduce new techniques which would have wide (or even slightly wide) use in math. Look at some of the Fields metal papers (e.g. restricted Burnside problem - Zelmanov - 1994 metal) and tell me how they changed mathematics.
For influences on math, consider Dirac (crazy British scientist who predicted to existence of the positron) whose ideas led L. Schwartz, L. to write "Théorie des distributions. Tome I,II"; distribution theory has had a huge influence on analysis.

Re:Already failed once!

And the first attempt of Andrew Wiles to prove Fermat's Last Theorem also failed, but he managed to patch it.