Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Motivation

In the analogy between prime numbers and knots, the prime number is thought sometimes as the circle of length $l([p]) = \text{log}\,p$. This is so you can express the zeta function as $$ \zeta(s) = \sum_{D\ge0} e^{-l(D)s}$$

where the sum goes over effective divisors on $\text{Spec}\,\mathbb Z$ and length is extended there by additivity. Similarly, you can do it to rewrite Dedekind zeta function for other number fields.

Question

I wonder, what is the right analogue of above formula for a manifold with metric? Perhaps:

  1. integration over all closed curves of the expression $e^{-l(D)s}$
  2. summation over positive sums of classes of closed geodesics.

I think I've heard something about definition 2, but I suspect if the two above are defined correctly they will be the same. Is it possible to formalize this definition? Do different formalizations lead to the same zeta-function?

Updates

Yes, I think this should be related to Laplacians, Selberg trace formula and dynamical system zetas. What I said I've heard about definition 2 was probably the Selberg zeta, but I can't say it clearly, hence questions.

share|improve this question
    
That sum looks a little bit like the McShane identity. en.wikipedia.org/wiki/McShane%27s_identity –  Sam Nead Jan 1 '10 at 20:59
    
Not exactly, because the sum there is alternating. But it should be in this terminology $L_T(-1, \chi) = 1/2$ where $L$ is a suitable analogue of $L$-function for torus $T$ and a "character" $\chi$. –  Ilya Nikokoshev Jan 1 '10 at 21:09
    
Does not really qualify for an answer, but perhaps the zeta-function for a Laplacian could be a good strating point? see e.g. here: books.google.com/books?id=gzJ6Vn0y7XQC&pg=PA144 –  mathphysicist Jan 2 '10 at 4:00
    
I think that zeta-function of a Laplacian should be related, thanks. –  Ilya Nikokoshev Jan 2 '10 at 10:32

2 Answers 2

There's been a lot of work since Smale's idea of a dynamical zeta function for general flows (in particular geodesic flows). A good starting point would be this 12 year old review by Baladi. There is a large and more recent literature but I'm no expert, although this other review by Liverani and Tsuji is probably not far from current knowledge.

There's also a whole branch of physics around those ideas, indeed related to the spectrum of the Laplacian and applications to quantum physics and statistical physics. This nice physics book is a good start (in particular if you read the quote of Smale at page 3 of this chapter, and then remark 19.2 at the very end of that chapter you'll get a quick sense of the stuff you've aked for).

share|improve this answer

In a slightly different spirit, but still carrying the analogy between closed curves and primes, The Selberg trace formula relates a sum over the lengths of closed geodesics on a hyperbolic surface (compact or no) to the spectrum of the Laplacian on said surface. Googling `Selberg trace wiki' will get you started here.

This formula has provided inspiration for Guillemin and collaborators working in spectral geometry and micro-local analysis.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.