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."
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.
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.
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).
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.
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.
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!
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
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.
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.
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.
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.
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.
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 < 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...)
by Anonymous Coward
on Wednesday June 09 2004, @06:11PM (#9382624)
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?
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.
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).
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.
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.
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.
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.
by Anonymous Coward
on Wednesday June 09 2004, @06:31PM (#9382788)
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
by Anonymous Coward
on Wednesday June 09 2004, @06:48PM (#9382909)
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.
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
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.)
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.
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.
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?
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.
by Anonymous Coward
on Wednesday June 09 2004, @06:33PM (#9382793)
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...".
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.
If there's one thing I know (Score:5, Funny)
homer simpson (Score:5, Funny)
Parent
Re:If there's one thing I know (Score:5, Interesting)
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
Parent
Re:If there's one thing I know (Score:5, Informative)
----------------
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.
Parent
Re:If there's one thing I know (Score:5, Informative)
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).
Parent
Apology (Score:5, Funny)
"We humbly apologize for the complete illegibility of this proof. The mathematician responsible has been sacked."
Re:Apology (Score:5, Funny)
"A Slashdotter has discovered a truly wonderful proof of the sacking of the mathematician responsible, but his bandwidth is too narrow to host it!"
Parent
WTF? Mods? (Score:5, Informative)
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?
Parent
Re:WTF? Mods? (Score:5, Funny)
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.
Parent
Re:Apology (Score:4, Insightful)
Parent
Re:Apology (Score:5, Interesting)
Parent
Re:Apology (Score:5, Interesting)
My favorite selection:
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!
Parent
Re:Apology (Score:5, Interesting)
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
Parent
Good job (Score:5, Funny)
Re:Good job (Score:5, Insightful)
Parent
Gotta prove 'em all (Score:5, Funny)
Riemann-chu, I prove you! Then bust out the paper.
Re:Gotta prove 'em all (Score:5, Funny)
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.
Parent
Nope! Nice try (Score:5, Funny)
For some suggested approaches, see (Score:5, Interesting)
Parent
Failed proof (Score:5, Informative)
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! (Score:5, Funny)
Hilbert Turns in his Grave? (Score:5, Interesting)
"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? (Score:5, Interesting)
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
Parent
Re:Hilbert Turns in his Grave? (Score:5, Funny)
Oh yeah? Mine would be "Is Doom 3 out yet?"
Honestly, which is more likely?
Parent
I'm in trouble (Score:5, Funny)
The question, explained (Score:5, Informative)
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.
Parent
Oh Hocky Sticks!!!! (Score:5, Insightful)
Thanks!
Parent
Re:The question, explained (Score:5, Informative)
Great Cthulhu help me, but I'm going to try and answer this for you.
We have natural numbers - 1,2,3,
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,
And yet, the discovery (or creation
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 < 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:
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Is it... (Score:5, Funny)
Impact on crypto? (Score:4, Interesting)
Re:Impact on crypto? (Score:5, Informative)
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.
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Re:Impact on crypto? (Score:5, Informative)
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).
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Re:Impact on crypto? (Score:5, Informative)
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.
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Hm (Score:5, Funny)
what?
Seems not-unlikely to be wrong (Score:5, Informative)
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 (Score:5, Informative)
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Re:Seems not-unlikely to be wrong (Score:5, Informative)
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.
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The Problem (Score:5, Informative)
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
actual paper (Score:5, Informative)
So if his "proof" isn't obviously wrong, it'll likely take quite a while for the experts to verify.
Much ado about nothing? (Score:5, Informative)
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
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
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Cool background material (Score:5, Informative)
If nothing else check out the animation [ex.ac.uk].
mind-boggling
Why people haven't believed him so far (Score:5, Informative)
The reasons why most specialists doubt that his approach can ever yield the result are well described in this paper [arxiv.org] from 1998:
(i.e., despite the name, the "generalized RH" proved by de Branges actually did not include the standard RH as a special case.)Re:Proof of theory (Score:5, Insightful)
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:
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.
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Re:Proof of theory (Score:5, Funny)
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.
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Re:Already failed once! (Score:5, Insightful)
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?
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re:Already failed once (Score:5, Insightful)
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.
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Re:What are the consequences for cryptography? (Score:5, Informative)
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.
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Re:quick google search (Score:5, Insightful)
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Re:de Branges' reputation with other mathematician (Score:5, Funny)
He appears to be 72.
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Re:Died before he could prove it (Score:5, Funny)
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