Comments on: Riemann hypothesis proved ?

By: Walt Pohl

Walt Pohl — Fri, 11 Jun 2004 19:07:31 +0000

Jeremy: I’m inclined to think the twin primes conjecture is true, but the answer is interesting whether true or false. What’s surprising is that we don’t know the answer either way.

By: Steve

Steve — Fri, 11 Jun 2004 17:45:39 +0000

No, primes are pretty common; much more common, for instance, then perfect squares or cubes, but on the other hand much less common than odd numbers. And, in fact, one of the side benefits of proving the Reimann Hypothesis would be that the prime number theorem could be calculated much more exactly. That's one of the reasons it's considered such an important question -- it ties into an big unanswered question in a different branch of math. (Demonstrating that P = NP or P != NP would also be a hugely important result, but it's less pure-math-y in my mind, and the consequences would certainly be much more directly applicable to the real world via algorithm writing.)

By: Anarch

Anarch — Fri, 11 Jun 2004 15:09:33 +0000

Jeremy’s intuition is correct in that the sum of the reciprocal twin primes converges—even if there are infinitely many—while the sum of the reciprocal primes diverges. [It’s Brun’s theorem (?), and I don’t pretend to know anything about the proof.] The catch is that “being infinite” isn’t even close to questions of sparsity or density; it’s one of the more obvious ways to concretely introduce someone to the concept that “infinity is REALLY big!” As to this: I’m not sure the Poincare conjecture is definitely considered to be settled. Last I heard, people were still digesting the announced proof. The proof they’re digesting, by a Russian whose name I can’t remember atm, is actually of the full Geometrization Conjecture and not merely the Poincare Conjecture. It’s entirely possible the full GC will fail but the PC will go through; again, though I’m not qualified to comment.

By: J. Ellenberg

J. Ellenberg — Fri, 11 Jun 2004 13:48:26 +0000

Jeremy, No, primes are pretty common; much more common, for instance, then perfect squares or cubes, but on the other hand much less common than odd numbers. If primes were distributed “randomly” throughout the integers, one would certainly expect infinitely many twin primes. Interestingly, the Green-Tao theorem alluded to above (caveat: I haven’t read the paper so only know about the argument by hearsay) gives a precise sense to the assertion “the primes are distributed in an approximately random way” and shows that ANY sequence of integers which is “approximately random” in their sense contains arbitrarily long arithmetic progressions. In some sense, their theorem is not really a theorem about primes at all.

By: Abiola Lapite

Abiola Lapite — Fri, 11 Jun 2004 11:52:25 +0000

——-BEGIN PGP SIGNED MESSAGE——- Hash: SHA1 “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.” The “apology” is not the proposed proof. The actual proof is the innocuously titled Riemann Zeta Functions”, which stretches on for 124 pages. That said, I still don’t think there’s more than a miniscule chance de Branges has nailed the Riemann Hypothesis, as he’s been announcing flawed “proofs” of it for years now.——-BEGIN PGP SIGNATURE——- Version: GnuPG v1.2.4 (MingW32) – GPGshell v3.10 Comment: My Public Key is at the following URL: Comment: http://www.alapite.net/pgp/AbiolaLapite.txt iD8DBQFAyY6/OgWD1ZKzuwkRAh4DAJ9usOSaHBy+sYHehwvrvbxnrfsbnACfWYKB D5+1exA6+2FfJVuwfbeNnSU= =DZjM——-END PGP SIGNATURE——-

By: Jeremy Osner

Jeremy Osner — Fri, 11 Jun 2004 10:37:42 +0000

Walt -- so there should be infinitely many numbers x for which both x and x + 2 are prime? This seems intuitively dubious. (Which means nothing, but) -- I always think of primes greater than 7 as being quite sparse.

By: mat

mat — Fri, 11 Jun 2004 10:05:14 +0000

The De Branges not-quite-a-proof paper was quite funny as well at the end, full of grandiloquent observations about how this proof was a delicious finding because of my family motto being blablabla… etc.

By: Walt Pohl

Walt Pohl — Fri, 11 Jun 2004 08:27:21 +0000

I’m not sure the Poincare conjecture is definitely considered to be settled. Last I heard, people were still digesting the announced proof. De Branges has a history of announcing proofs of results that later turn out to be incorrect, so who knows. Twin primes and Goldbach are by themselves not that important, but they are typical examples of whole classes of questions that we have no idea how to answer. Let’s put the twin primes conjecture another way: Consider the polynomial x^{2 + 2x. Now, value of this polynomial is never prime (for natural numbers), since it factors as x(x+2). How often is it a product of two primes? If the twin primes conjecture is true, then it’s infinitely often.

There’s nothing special about x}2 + 2x, though. We can consider any polynomial, and ask how often does it take on prime values, the product of two primes, etc. For nonlinear polynomials, we don’t know the answer. You can rule out certain cases happening by elementary algebra, but beyond that, we don’t know. If we knew the answer for any one, such as twin primes, we would probably know the answer for a bunch of others.

By: Chris Bertram

Chris Bertram — Fri, 11 Jun 2004 07:53:54 +0000

I note that the same Mathworld page also announces Ben Green and Terence Tao’s proof of the existence of arbitrarily long progressions of primes. Green is a Bristol boy and recently gave a presentation of this in our Maths department here. My (more mathematically savvy than I) colleagues who attended came away really impressed and haven’t stopped talking about it for weeks.

By: Andrew Boucher

Andrew Boucher — Fri, 11 Jun 2004 06:16:37 +0000

There are frequently claims of proofs of important theorems. Usually they don’t pan out. The Riemann Hypothesis is not the last significant mathematical conjecture. It is probably considered the most important open mathematical problem, so if it were solved, there would have to be a new one. P = NP (what one comment refered to as the Travelling Salesman problem) would probably be a candidate, although there are also many simply stated problems still open in number theory. For those who want to get wealthy, there are the seven problems set by the Clay Mathematical Institute (RH and P = NP are two of them). As another poster has pointed out, the recent (serious) proof of the infinitude of the twin primes apparently has a problem.

By: Kieran Healy

Kieran Healy — Fri, 11 Jun 2004 06:06:30 +0000

The way Mathworld put it just made me laugh out loud: bq. 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.

By: Giles

Giles — Fri, 11 Jun 2004 05:44:08 +0000

I thought that the travelling Salesman problem is still out standing – its a mathematical problem that will also be very useful

By: Aaron Bergman

Aaron Bergman — Fri, 11 Jun 2004 05:34:06 +0000

The twin prime proof has been withdrawn.

By: Michael Kremer

Michael Kremer — Fri, 11 Jun 2004 04:50:53 +0000

I now see that the Mathworld article linked to in the original post also points out the likely error but is optimistic that it might be fixable. See the last paragraph of the report.

By: Kevin Drum

Kevin Drum — Fri, 11 Jun 2004 04:50:05 +0000

De Branges has been claiming to have a proof of Riemann on a periodic basis for a very long time. Treat this one with a very big grain of salt for now.