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."

My own personal proof

Non-trivial zeroes of the zeta function are 1/2 because they naturally form as wholes, but as we all know a grue can't resist the tasty flesh of a non-trivial zero. I posit that the only way to prove the hypothesis is to kill a grue and vivisect it to search for the other half of the non-trivial zero. So until someone is brave enough to fight a grue and extract the flesh of the non-trivial zero, that million dollars is going unclaimed.

Re:

...as we all know a grue can't resist the tasty flesh of a non-trivial zero.

True enough. The interesting nature of non-trivial solutions is apparent to all; grue and non-grue alike.

I posit that the only way to prove the hypothesis is to kill a grue and vivisect it...

But not in that order! "Vivisection" means dissection while alive. You'd need to capture a live and viable grue and then not kill it until (too early in the) dissection.

So until someone is brave enough to fight a grue and extract the flesh of the non-trivial zero, that million dollars is going unclaimed.

Mathematicians are known to go adventuring from time to time, but mostly they seem to prefer coffee or tea for the extraction of proofs from the darkness.

Re:

Apparently I fail at reading comprehension. I blame public schooling.

I have already solved this!

...But unfortunately I do not have
enough room in the margin of this
text area to display it properly.

Re:I have already solved this!

am on ship on way to england stop have solved riemann hypothesis stop will give details on return stop

Re:I have already solved this!

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.

The Riemann Hypothesis, by Robert Ludlum

Cue the creepy, hushed voice-over:

In a University in Lower Saxony, a mathematician had formulated a remarkable conjecture. Its effects would be felt worldwide.

The Riemann Hypothesis, by Robert Ludlum. Now in paperback.

smile and nod

(just smile and nod, smile and nod. they'll never know you have no idea what this means)

Re:smile and nod

I can vouch for the smile-nod method.

I got through half a year of classes on game-theory in Korean with it. The professor never noticed I didn't speak a word Korean. Tests were in English, and the guy just kept asking ending in '...isn't that right, studentX?' or '...do you not agree, studentY?'.

Smile and nod baby, smile and nod. Best followed by a short single chuckle, as if the intrinsical irony of reality does not elude you.

By the end of the semester the guy actually seemed to like me.

What's really going on here

Booker and Ce Bian constructed certain degree 3 L-functions, but it is best to think of their discovery as follows: there is a complicated 5-dimensional membrane known to mathematicians as "SL(3,Z)\SL(3,R)/SO(3)". This membrane has subtle number-theoretic symmetries, so that its modes of vibration encode number-theoretic information. These modes (and their vibrational frequencies) are being extensively studied, but they are very transcendental objects so they cannot be written down explicitely and must be computed numerically. While certain modes (and frequencies) were already known numerically (they can be constructed from vibrational modes of the 2-dimensional membrane "SL(2,Z)\SL(2,R)/SO(2)" via something known as the Gelbart-Jacuqet lift) we now have the the first numerical computation of "native" modes of the 5-dimensional membrane -- those that aren't related to lower-dimensional cases. To each such mode of vibration there is an associated "L-function (similar to the Riemann zeta function), and it is the L-functions that were constructed. In fact, verifying that the approximate L-functions that were found correspond to actual modes of vibration is not easy (in the 2-dimensional case there is important work of Booker with others about this).

In short, this is an important advance in automorphic forms, but it is so technical that it doesn't belong on SlashDot.

It is important to realize that while indeed there is a ("Generalized") Riemann Hypothesis associated to these L-functions, numerically computing them represents zero progress toward proving the Riemann hypothesis for these L-functions or the original Hypothesis for the Riemann zeta function. At most this will allow very approximately computing some of their zeros and thus a weak check on the GRH for these L-functions.

Re:What's really going on here

Wow, it's so clear now!

Re:What's really going on here

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.

Re:

I sincerely tried to follow all that, but it's so far over my head that it's in orbit around Jupiter.

Re:What's really going on here

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?

Re:What's really going on here

Let's try:

• The "membranes", "modes" and "frequencies" here are already a physical analogy. Number theorists study objects (``automorphic forms'' -- no matter why they are called this way) that live on some ``manifolds'' (no matter what that means, either). But to get some intuition you can replace ''manifold'' with ''taut membrane'' (like a drum) and ''automorphic form'' with ''normal mode'' a.k.a. basic ''standing wave'', as you call it. An important problem in mathematical physics is to find what are the possible frequencies of standing waves on a particular surface. The problem here is analogous.
• To see a picture of the 2-dim membrane I was talking about, see here [springer.de]. Start by taking a half-infinite strip of width 1, and cut off a semi-circular bit at the bottom like in the picture (the strip extends infinitely far at the top. Next, glue the two infinite sides together so the strip becomes a cylinder. Finally (that's not in the picture) imagine that as you go further and further up the cylinder, its radius becomes smaller and smaller, so the real thing is a kind of infinite funnel.
• To see what a standing wave on this membrane looks like, see here [umn.edu] (this was computed numerically by Dennis Hejhal).
• The "lift" that takes a standing wave on this space to a standing wave on the 5-dim space is really complicated (and is a very indirect construction). There just isn't a non-technical way to describe it.
• However, we know what the "lift" does to the frequencies: if you start with a standing wave you found numerically, and approximately know its frequency, then you know there will be a lifted guy of a calculatable frequency on the 5-dim space. So the interesting problem is to find standing waves with frequencies which are different from the ones we already know about (because we have calculated a lot of standing waves on the 2-dim surface).
• One symmetry this infinite funnel has is left-right reflection (it is apparent both in the picture of the strip and in the picture of the vibrational mode). The other symmetries are difficult to describe in a blog post. What's important is that the modes of vibration must respect the symmetries.
• It is true that to each such ''standing wave'' (on the 2-dim surface, on the 5-dim space, and on others) there is an associated L-function. The Riemann Hypothesis for these L-function (the same formulation: all zeros are on the critical line) is called the "Generalized (or Grand) Riemann Hypothesis" or GRH.
• It was possible to calculate a few zeros of the newly-found modes, and see that indeed they are where they are supposed to be. This gives some evidence for the GRH. Calculations like this can always falsify the GRH (by finding a zero off the line). However, these calculations don't represent any progress toward proving the GRH -- that was confusion on part of the person who submitted the story to slashdot.

I hope this helps.

Re:What's really going on here

Paul Cohen. It was Paul Cohen [wikipedia.org].

And he didn't solve the continuum hypothesis. He showed that you cannot prove CH from the ZF axioms. Gödel had previously show that you cannot *disprove* CH from ZF (unless ZF is inconsistent). Together these results show that CH is independent of ZF.

So CH is still an unresolved problem today. As far as anyone knows, either CH or its negation can be taken as a separate axiom of itself, which leaves it an open question.

Unproven since 1859???

If they'd have left it alone in 1858 we wouldn't be having this trouble. If it ain't broke, don't fix it!

question

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?

Re:question

Now what the heck are degrees of those L-functions?

This is where things get technical. The Riemann Zeta-function $\zeta(s) = \sum_n n^{-s}$ has the Euler product representation $\zeta(s) = \prod_p \left( 1 - p^{-s}\right)^{-1}$. Similarly, the Dirichlet L-functions $L(s;\chi) = \sum_n \chi(n)/(n^s)$ have the Euler product $\prod_p L_p(s;\chi)$ with $L_p(s;\chi) = 1/( 1 - \chi(p)/(p^s))$. In both cases, the factor at each prime $p$ takes the form $1 / ( 1 - a(p)/p^s )$, for some number $a(p)$ depending on $p$. We think of this factor as a the inverse of a polynomial of degree 1 in the variable $p^(-s)$ (the polynomial is $P(T) = 1 - aT$).

Similarly, to GL(3) Hecke-Maass forms such as the ones computed by Booker and Ce Bian, there is an attached L-function $L(s;f)$ which can be represented as an Euler product, $\prod_p L_p(s;f)$. This time, however, the local factors $L_p(s;f)$ are the inverses of cubic polynomials, that is $1/L_p(s;f)$ takes the form $P(p^-s)$ where $P(T) = 1 - aT - bT^2 - cT^3$ for some coefficients $a,b,c$ depending on $p$ (and on $f$, of course). This is why we call it an L-function (or Euler product) of degree 3.

Using the Fundamental Theorem of Algebra, it is common to factor the polynomial $P(T)$, and write it in the form $\prod_{j=1}^{3} ( 1 - \alpha_j(p) T)$. Thus an Euler product of degree $d$ takes the form:

\prod_p \prod_{j=1}^{d} 1/(1-\alpha_j(p) p^{-s})

I forgot to credit Marginal Revolutions blog

Submitter here. Right after hitting submit, I realized I'd forgotten to link to marginal revolutions, an economics blog that pointed me to the story.
http://www.marginalrevolution.com/marginalrevolution/2008/03/assorted-link-4.html [marginalrevolution.com]
http://www.marginalrevolution.com/ [marginalrevolution.com]

Re:

Regardless of the money,
Ce Bian and Andrew Booker (and their computer) should at least win SOME prize and may need to practice their Sweedish.....

Re:wow...

Actually, the Riemann hypothesis is pretty important, given that a proof of it would tell us about the distribution of prime numbers, and prime numbers are the wheels which keep e-commerce turning (RSA anyone?) Also, concerning scientific results which sound like Robert Ludlum novels, my own personal favourite is the Born Approximation - the least popular in the Bourne series.

Do you know what you're talking about?

Actually, what the RH tells us about the distribution of prime numbers is be pretty useless regarding RSA. To get anywhere you need the Extended Riemann Hypothesis (covering Dirichlet L-functions) and even the full force of the "Generalized Riemann Hypothesis" (covering all automorphic L-functions) is not known to help with the really important problem here -- factoring.

Re:

That show is the best mathy/sciency show on television, mostly because they never, ever get the science wrong. Also, there's some good acting.

You miss the point

infiltrating hundreds of thousands of computers to work on the solution

The solution isn't to be found through massive computing effort. They are looking for a proof, not a computation. They need creativity, not horsepower.