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."

Comments

Failed proof

[MobyDisk]

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.

WTF? Mods?

[Unnngh!]

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?

The Problem

[Anonymous Coward]

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

Re:What are the consequences for cryptography?

[Anonymous Coward]

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.

Re:Impact on crypto?

[susano_otter]

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.

actual paper

[Anonymous Coward]

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.

Re:Seems not-unlikely to be wrong

[roll_w.it]

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

[mrthoughtful]

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.

Cool background material

[Anonymous Coward]

A cool overview of why this is such an interesting hypothesis.

If nothing else check out the animation.

mind-boggling

The question, explained

[SamSim]

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.

Re:The question, explained

[h4rm0ny]

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

Re:If there's one thing I know

[PeeCee]

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).

Why people haven't believed him so far

[This is outrageous!]

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 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.)

Much ado about nothing?

[Scorillo47]

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