String Theory & Riemann Hypothesis: Is There a Connection?

[pivoxa15]

Could the maths of string theory or versions of it lead insight into the Riemann hypothesis as, for a start both are about mathematics in the complex plane.

Anyone working on this connection at the moment?

 

[matt grime]

for a start both are about mathematics in the complex plane.

That probably encompasses 50% of mathematics, picking a number at random that seems plausible.

 

[Gib Z]

I'm most 100% sure but I think there are results about RH where it shows that it is true for some upper bound, and all the requires is that the lower bound is verified. However the only problem is that the upper bound is unimaginably huge, current supercomputers could spend millenia verifying those cases. What I say is, whilst mathematicians try to prove the theorem in a more traditional method, computer scientists and physicists should work on increasing computer speed (quantum computing would definitely help). Then perhaps with another few hundred years of computer techonology, a supercomputer could verify the theorem in less an a century? Those are my musings on the subject anyway =]

 

[CRGreathouse]

I'm most 100% sure but I think there are results about RH where it shows that it is true for some upper bound, and all the requires is that the lower bound is verified. However the only problem is that the upper bound is unimaginably huge, current supercomputers could spend millenia verifying those cases.

Really? If you find a cite I'd love to see it.

 

[Gib Z]

Damn it! Obviously I'm making this up which I wish were true =]

 

[sepowens]

You are missing the point

I think you are missing the point of a proof. It doesn't matter how far 'up or down' computers show that Riemann holds, they can never prove the theory. While it only takes one counterexample to disprove a hypothesis(A point meeting the criteria specified in the RH which is not on the 1/2 line in this case) you can never prove it by counting examples of where it holds on an infinite set.

If there was an upper bound then we could probably put it to bed already as there has been no counterexample up to an absurdly high value of empirical data showing it holds.

I know it seems nit-picky but the concept of proof is what distinguishes mathematics from all of the sciences and even the vagaries of life. Evolution, quantum mechanics, relativity and all of the other cornerstones of our understanding of life, the universe, and all that are theories. This only means we have a sizeable body of evidence that shows we might be right and nobody has proved us wrong. So far there is not one theory of science that is concrete, indisputable and not open to revision by some bright spark working aas a patent clerk tomorrow. It is all just our best guess so far.

In mathematics we do not have theories but magnificent theorems. These great wonders will stand for all time in every corner of the universe and even in other universes. They can not be argued with or changed. That is the true beauty of mathematics.

But keep thinking about it as that is how great problems like this get solved

 

[eigenzeitt]

In mathematics we do not have theories but magnificent theorems. These great wonders will stand for all time in every corner of the universe and even in other universes. They can not be argued with or changed. That is the true beauty of mathematics.

Oh, no. Unfortunately, in mathematics we have a set of axioms. These axioms as Godel proved a long ago are undecidable as to their truth or not. Thus all theorems ( theories from greek theoro=inspect ) which are logical conclusions of axioms can be refuted, changed, argued, whatever...
If I prove to the scientific community with a childishly ridiculously easy but ingenious counter-example as I have done here in this site that the whole of special relativity theory is grounded on probability premises thus it is not a classical theory, it is not the fault of the theory but of the axioms a classical theory is based upon. Don't you think?

 

[CRGreathouse]

If there was an upper bound then we could probably put it to bed already as there has been no counterexample up to an absurdly high value of empirical data showing it holds.

I don't know about that. If there was an upper bound of Skewes' number we wouldn't really be any closer to solving the problem, as we'll never be able to test that far directly:

Consider a quantum supercomputer with a googol particles (more than in the whole universe) calculating for a googol seconds (longer than the lifetime of the universe) doing a googol operations per second (faster than any plausible computer). We'll give it
$$2^{10^{100}}$$
effective operations per quantum operation since it gets the whole 'quantum speedup' thing. (Yes, this is optimistic.) So the overall speed is
$$10^{200}\cdot2^{10^{100}}<e^{e^{230}}<e^{e^{e^{79}}}$$

So even with an impossibly fast quantum supercomputer larger than the universe, a googol seconds aren't long enough to check that bound, even with an atomic 'check the next Riemann zero' function.

I think you are missing the point of a proof. It doesn't matter how far 'up or down' computers show that Riemann holds, they can never prove the theory.

Sure it does. Lots of calculations are based on the RH holding up to a particular height. Pierre Dusart's excellent bounds on $\pi(x)$ is the first example that comes to mind, but there are many more.

 

[sepowens]

Good point on the axioms. Read Goedel carefully and you will find that for a given set of axioms that it is consistent. Now the choice of axioms is obviously problematic although it can lead to such things as the Euclidian, parabolic and hyperbolic geometry models. The real point is that for a given set of axioms the system will be consistent as Goedel himself pointed out. This leads us to a sureness that the proof will stand based only on the few axioms chosen. The problem that Goedel pointed out is that for any set of axioms that is consistent it can never be complete. That is there will always be questions which can not be answered with any choice of axioms. You can add axioms and reduce this set but it is infinite.

As to other hypothesis depending on the RH holding up to a certain point, it proves nothing as to the validity of the RH itself. It only shows that the criteria for RH are not violated up to the point required by the other hypothesis. They are built on the RH not the other way around. Proving it holds up to the point they need it to do does not mean that it can not fail on the next interval chosen.

Point well taken on Skewes and I will concede that one gladly. Very foolish statement on my part. I suppose I had it in my mind that if we established a definite bound for Riemann that it would be calculable under the given level of technology that established it and terefore within the realm of the current or near future test data. Chose the wrong axiom...