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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.

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  • $\begingroup$ That sum looks a little bit like the McShane identity. en.wikipedia.org/wiki/McShane%27s_identity $\endgroup$
    – Sam Nead
    Commented Jan 1, 2010 at 20:59
  • $\begingroup$ 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" $\chi$. $\endgroup$
    – Ilya Nikokoshev
    Commented Jan 1, 2010 at 21:09
  • $\begingroup$ 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 $\endgroup$
    – mathphysicist
    Commented Jan 2, 2010 at 4:00
  • $\begingroup$ I think that zeta-function of a Laplacian should be related, thanks. $\endgroup$
    – Ilya Nikokoshev
    Commented Jan 2, 2010 at 10:32

2 Answers 2

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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).

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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.

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