Slashdot Mirror
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."
It's that mathematicians love to exaggerate! Like infinity is infinite, or pi goes on forever! Those guys are always talking big.
Apology for the proof of the Riemann hypothesis (in pdf format).
"We humbly apologize for the complete illegibility of this proof. The mathematician responsible has been sacked."
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.
-I am an elective eunuch.
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.
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.
Ha! They've already found an error in the proof! All that he posted was his apology! :-)
Yes, I was actually confused at first. For the non-math geeks like myself, who are feeling stupid, look at definition 2a of apology.I don't want to give it away, but you'll see it.
"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
You know you're in trouble when you don't even understand the question.
When they came for the communists, I said "He's next door. Take him away. Goddam commies."
... 42?
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?
I think I speak for all non-mathematicians when I say:
what?
I am a filthy pirate.
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
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.
IAALS.
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.
Any sufficiently well-organized community is indistinguishable from Government.
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.
Boom Shanka
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.
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.
http://www.maths.uwa.edu.au/~berwin/humour/invalid .proofs.html
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.
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.
This comment was written with the intention to opt out of advertising.
mmmmmmmm......infinite pie..!
A cool overview of why this is such an interesting hypothesis.
If nothing else check out the animation.
mind-boggling
First: complex numbers, 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. 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. 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 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 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 suggests that they are... but until today nobody has been able to prove it.
qntm.org
The reasons why most specialists doubt that his approach can ever yield the result are well described in this paper 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.)This is...
O
U
T
R
A
G
E
O
U
S
!
There's the occasional post that deserves to be modded to "+10 -- Best Damn Thing I've Read On Slashdot This Year".
Thanks!
You are not a beautiful or unique snowflake -- but you could be if you got off your ass.
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.
/. until now :-)
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
Don't try to use the force. Do or do not, there is no try.