Mathematician claims proof of Riemann Hypothesis

I was going to write up something on this, but Chad Boutin did a great job summarizing the claim:

WEST LAFAYETTE, Ind. � A Purdue University mathematician claims to have proven the Riemann hypothesis, often dubbed the greatest unsolved problem in mathematics.

Louis De Branges de Bourcia, or de Branges (de BRONZH) as he prefers to be called, has posted a 124-page paper detailing his attempt at a proof on his university Web page. 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 whoever accomplishes it first � has encouraged de Branges to announce his work as soon as it was completed.

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De Branges is perhaps best known for solving another trenchant problem in mathematics, the Bieberbach conjecture, about 20 years ago. Since then, he has occupied himself to a large extent with the Riemann hypothesis and has attempted its proof several times. His latest efforts have neither been peer reviewed nor accepted for publication, but Leonard Lip#z, head of Purdue's mathematics department, said that de Branges' claim should be taken seriously.

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Complete Story

I hope peer review turns out positive! This could be an amazing revelation for mathematics. MathWorld has an excellent discussion on the Riemann Hypothesis. Interestingly enough, it mentions DeBranges' proof. This just came out, so the people at MathWorld must really be on top of things, unless they're talking about one of his earlier attempts. After all, he has been working on Riemann for the last 20 years or so. In either case, it's good to see that mathematics isn't dead, as some would claim. It's also refreshing to see something other than "largest prime number found" in the news.

Edit: Looking back, I've noticed this isn't quite as new as I thought. His paper was made available two weeks ago, and MathWorld does refer to this latest paper. Guess we'll just have to see what his peers think.

[edit on 6/13/2004 by PurdueNuc]

It would be great to see the Riemann hypothesis proven. It is really going fast these last few years proving great problems: Fermat, Twin Primes coming very close. MathWorld is not very positive about this proof though:

Riemann Hypothesis "Proof" Much Ado About Nothing

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.

Interesting that Mathworld would be so negative early on. I expect more froma publication that relies on solid research and evaluation. I really hope he has proved it!

whats the riemann hypothesis?

The hypothesis is that all solutions of z(s)=0 lie on a line in the complex plane (source):

The Riemann hypothesis asserts that all interesting solutions of the equation

z(s) = 0

lie on a straight line. This has been checked for the first 1,500,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers.

The Riemann zeta-function z(s) can be expressed in a different ways (more information), but my favourite is the simplest one: