Talk:Riemann hypothesis

Nash

Can someone comment Nash's words about zeroes of Euler - Riemann zeta function ζ(s) that its zeroes are singularities of space and time. This one is of course from Howard's film Beautiful Mind.
XJam [2002.03.24] 0 Sunday (0)

Analytic continuation

Sorry to be so dumb, but I never understood how -2, -4 etc can be zeroes of the function. Why isn't zeta(-2) just the sum of squares 1+4+9+... ? [2002.08.28] Stuart Presnell

Because obviously that would not work ! zeta(s) is defined using (sum 1/z^s) over all complex numbers s=x+iy with x > 1, then extended to the whole complex plane (excepted at -1) using analytic continuation. That is, zeta is the unique analytic function (= holomorphic function) on the complex plane (less -1) that matches the sum where it is defined. That is described in the first paragraph of the article on the Riemann zeta function. See also "analytic continuation". -- FvdP Sep 5 & 7, 2002

Ah, thanks! I was relying on the little knowledge I had gleaned from a few popularisations - didn't know about the 'analytic continuation' aspect of it. If only I had bothered to read the appropriate Wikipedia entry... [2002.09.13] Stuart

Divergence and meaning

Please forgive my ignorance, since I am not really into mathematics (in fact only 13 years old), but don't all inputs less than or equal to 1 in the zeta function cause it to be equal to infinity? Ilyanep 14:37, 9 Jun 2004 (UTC)

No - the infinite series is actually meaningless in that region; there are some other ways of representing the function, which allow one to discuss it there.

Charles Matthews 15:30, 9 Jun 2004 (UTC)

De Branges (I)

An interesting looking article: [1] - Someone claims to have proven the Riemann Hypothesis. I picked this up off of /. - Xgkkp 23:25, 9 Jun 2004 (UTC)

De Branges - many people's hearts might sink. Of course the Bieberbach conjecture business counts in his favour; but still. Charles Matthews 07:20, 10 Jun 2004 (UTC)

Can anyone explain why anyone cares?

We state this is an important unsolved problem, but I have no idea either before or after reading the article why it's important. What makes this problem important? Yes, I know it will probably take about 3 pages to say why, and that it will have to be grossly oversimplified, but it would be nice to have some idea of what it would mean if [1] it were proved true or [2] it were proved false. - Nunh-huh 02:34, 11 Jun 2004 (UTC)

What, for practical applications? Not any, yet. But, prime numbers are *extremely* important in cryptography, and this theoream is extremely important in prime number theory. →Raul654 02:49, Jun 11, 2004 (UTC)

There was a lengthy explaination in a book called The Music of the Primes (dunno if that's an article), which says that if the Riemann Hypothesis is false, then that means that the prime numbers have a certain order to them. It would mean that the 'prime number coin' is biased and doesn't have a probability of 'landing' on a prime 1/log(n) (to base e) times, that there is an order. And if there is an order to the primes, it kind of jeopardizes some encryption methods (such as RSA, a method based on the difficulties of factoring numbers) Ilyanep 02:56, 11 Jun 2004 (UTC)

Ah. For a non-mathematician, this is not "intrinsically obvious". I hope you'll add the explanation to the article. - Nunh-huh 03:01, 11 Jun 2004 (UTC)

I think crypto is not directly relevant - though it is constantly brought into discussions of number theory. It is more like this: what we can know in prime number theory depends on the extent to which there can be a 'conspiracy' amongst prime numbers (it would have to be very large prime numbers, another reason why crypto misses the point) which defeats the kind of reasoning that says they are entities largely independent of each other. RH is probably the deepest single, simple statement saying 'no conspiracy'.

Charles Matthews 05:46, 11 Jun 2004 (UTC)

About Applications - the prime number theorem is true; what is at issue is whether the error term is random walk-like (square root of the main term, or nearly) or bigger.

Charles Matthews 16:37, 11 Jun 2004 (UTC)

Okay, let me go fix that Ilyanep 16:42, 11 Jun 2004 (UTC)

External link

I have cut out this one ("WhatPC?" Article Article on how proof of the Reimann hypothesis could destroy E-Commerce). I think it has no useful content.

Charles Matthews 19:29, 7 Sep 2004 (UTC)

Agreed. No counter examples have ever been found even after much searching. It's proof would just validate what everyone thinks is probably true about primes. It's disproof might jeopardize cryptography. pstudier 21:56, 7 Sep 2004 (UTC)

And it might not have anything much to do with crypto, in fact. Charles Matthews 19:15, 20 Oct 2004 (UTC)

500 or 1000?

just a minor point, from the book "music of the primes" the quote from hilbert is waking up after "five hundred years" not a thousand. Searching on google seems to give conflicting answers though, anyone read the actual article ?.

Wombat 02:04, Dec 8, 2004 (UTC)

De Branges rumour

I'm moving this off the page. Nothing recent or newsworthy, I think. Charles Matthews 12:13, 12 Feb 2005 (UTC)

A possible proof of the Riemann hypothesis

In June 2004, Louis De Branges de Bourcia of Purdue University, the same mathematician who solved the Bieberbach conjecture, claimed to have proved the Riemann hypothesis in an "Apology for the proof of the Riemann Hypothesis"[2](pdf). His proof will soon be subjected to review by other mathematicians. De Branges de Bourcia has announced a proof a number of times, but all of his previous attempts at this proof have failed.

The full purported proof is "Riemann Zeta functions" [3](pdf).

The proof's method has been tried before unsuccessfully. Linked is Conrey and Li's counterexample on the problems in the earlier version of his proof. [4] The example involves a numerical calculation. The authors also give a non-numerical counterexample, due to Peter Sarnak. On the other hand, De Branges's successful proof of the Bieberbach conjecture was also preceded by his failed proofs of it.