Wednesday, July 9, 2008

The Riemann hypothesis - why does it matter?

Mathematicians regard the Riemann hypothesis as very important, not least because a fair number of important theorems have been shown, given the assumption that the Riemann hypothesis is correct, so as soon as it's shown to be true, a whole pile of other interesting stuff is also true. Its impact is both practical and theoretical.

But when I was talking to William earlier today, he summed it up well:
"If it isn't true, math is a lot weirder than we think it is."

[Oh, and in case you are wondering what the situation with Li's proof is - while I was away on vacation for a few days, the paper was withdrawn. It appears Li could not rescue the proof. ]


Dana Hunter said...

From what I remember of math, not being able to rescue the proof means proof go poof. Too bad - I may not know diddly squat about this stuff, but I always like to see some clever buggers figure out proofs that have eluded others for hundreds of years. And this one seems far more important than some.

How was vacation?

Efrique said...

Yes, the proof is dead.

It had an error as pointed out by the commenters I quoted earlier.

At that point, either the proof is fixed (the flaw can be overcome by making some additional argument, as happened for example with Wiles' famous proof, which had flaws) or it isn't.

Li made several attempts to fix it, then withdrew the paper because he couldn't. So yes, "couldn't rescue it" means that there's currently no proof.

With the big ones, this might happen several time before we get there (assuming we even can*).

* However, I will say that I believe that the Riemann hypothesis will eventually be proved. But it looks like it will need more powerful tools than we have right now, or some incredibly clever insight that we currently don't have. I don't think the Riemann hypothesis is in the class of true but unproveable statments.

[That last link is to a lego comic where the Mythbusters gain omniscience ... and realize that the Riemann hypothesis is true but unproveable. If you want to know *why* they're being ferried across the river Mnemosyne, you have to read the preceding comics. N.B. Do read the (lengthy) explanation of the Riemann hypothesis under the comic.]

valhar2000 said...

In one episode of Numb3rs, a mathematician announced that eh ahd proved Riemann's Hypothesis. People were all a-twitter because they realized that if you can prove the Riemann Hypothesis is true, you can easily develop an algorithm that alows you to crack most modern encryption schemes.

Thus, the bad guy kidnapped the guys daughter and tried to force him to give them the algorithm so they could use it to make investments with insider info and rob banks.

That said, my borther, who knows more math than I do, told me that could not be, because cryptographer's have already attempted to develop such cracking algorithms by simply assuiming that Riemann's hypothesis is true, and working from there. Thus, if you prove the Riemann Hypothesis, you validate work that cannot give you the desired result in the first place.

Is this correct?

Efrique said...

First, that's far from the only thing Riemann Hypothesis being true would potentially give you (assuming it could give you that at all).

Second, Numb3rs is fiction. I'm somewhat familiar with a number of the techniques they mention on the show (say a quarter to a sixth of them) and in the cases I am most familiar with, what you're getting in the show is basically a mention of a bit of mathematical or statistical technology that's somewhat connected to the problem at hand so that they can throw a bit of jargon around, give a vague (and often misleading) analogy and pretend it works like they suggest.

The point is to generate an interesting (and often vaguely plausible) story, so don't take what's going on as gospel. I have taken to interpreting the terms they use as "placeholders" for something else that would work more like what they're trying to do, because otherwise I keep dropping out of involvement in the story (because my brain keeps leaning over the back of the lounge and kibitzing with "it can't work like that, you know").

So, anyway, with all that in mind, think about this:

What went on in that episode of Numb3rs could be plausible IF the techniques in the proof itself gives important clues as to how to do the factoring of large primes required for breaking typical kinds of encryption.
(Some kinds of proof would not be helpful, some would be helpful. Imagine the proof in the show was of the second kind.)

tai said...

I prove Riemann Hypothesis.

Please see this.