Slashdot Mirror
A Step Towards Proving the Riemann Hypothesis
arbitraryaardvark writes "A new mathematical object has been discovered by Bristol University student Ce Bian. The Riemann hypothesis, unproven since 1859, has to do with the distribution of primes and something called L-functions. Bian has demonstrated the first known third-degree transcendental L-function. This apparently opens up a new way to go about looking for proofs of the Riemann hypothesis. There is an unclaimed $1 million prize for a valid proof. We've discussed a couple of earlier attempts to claim the prize."
Riemann zeta function is the "mother of all L-functions".
...
zeta(s)=sum(n=1, inf)(1*n^-s)
Dirichle L-function is defined as
L(f, s)=sum(n=1, inf)(f(n)*n^-s)
so when f(n)=1, Dirichle L-function becomes Riemann zeta function.
L-function is just another representation (called Euler product) of Dirichle L-function.
L(f, s)=prod(prime p=1, inf) P(p, s)
where
P(p,s)= 1 + f(p)p^-s + f(p^2)p^-2s +
The Euler product I figured must work similar to the usual prime number decomposition: you got the sum of 1's and you got a product of primes.
That is how far I got.
Now what the heck are degrees of those L-functions?
I do not believe in karma. "Funny"=-6. Do good and forbid evil. Yours, Oft-Offtopic Flamebaiting Troll.
Everybody have already noticed that most recent proofs of outstanding hardcore die hard theorems are mind-bogglingly long or simply numeric (it's not a proof, I know).
Does it mean we are closing to the Goedel's incompleteness levels of the development of formal number theory?
I do not believe in karma. "Funny"=-6. Do good and forbid evil. Yours, Oft-Offtopic Flamebaiting Troll.
You know, there's a lot of speculation about that. I suspect he did have a proof, but I'm skeptical that it was correct. There's no doubt that the man was brilliant but we've had people working on that question ever since Fermat died and no one has been able to produce a "simple, elegant" proof. (Fermat's own description, there.) But there's plenty of precedent for mathematicians making things inordinately complex before some young genius comes along and shows a magnificently simple way of achieving the same thing.
(just smile and nod, smile and nod. they'll never know you have no idea what this means)
Agree. I'm the guy who submitted the article, and I have no idea what it's about.
It just felt slashdotty.
Somewhat, but the parallel conjecture that went all the way back to Euclid couldn't be proven, even though it seemed largely true. Eventually Riemannian geometry arose as something that broke this well established conjecture. Often, yes, it's useful to assume conjectures, but don't underestimate the value of a proof, or even the value of failed proofs.
Umm... not to boost your ego too much, but if that seemed intuitive, you might consider reading ahead in your text books and looking at what else seems intuitive. If you manage to run your intuition through 3 var calculus, go for point set topology next. This "intuition" you speak of might be more of a gift than you realize. If you get through topology on your own, mention it to a professor who is known to be good at explaining things (those are usually the ones who actually understand and LIKE to explain things). He'll tell you what's next. You might have the ability to naturally translate between geometrical and symbolic view. Believe me, it's not common. And things you can learn to do with it are pretty cool.
Any guest worker system is indistinguishable from indentured servitude.
This article is related to Sage ( http://www.sagemath.org/ ), a free open-source math project. The article is about a computation (not using Sage) of an L-function, a computation about that L-function (using Sage), and a major new NSF-funded initiative to compute large tables of modular forms and L-functions that William Stein (director of the Sage project) is co-directing, which will have a large impact on Sage development.
Quite the contrary, actually - I think we need more discussions (and more posts) like this on Slashdot. It's a good starting point to look things up.
I already know a few things about L-functions and GRH, but I'm not sure what the "membranes" you refer to are. Are you speaking of the same "membranes" that appear in M-theory?
Yes, indeed. But of course I was joking. Also: This way of "fooling myself" usually maeks me go home and read the book for real, just to know what the hell I was talking about. :)
The "effort trying to fake it" somehow always ends up with me learning something...