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I'm trying to learn about the Selberg zeta function, but it seems like introductory texts assume more knowledge of Riemannian geometry than I'm comfortable with.

I have some basic questions that someone might be able to help with:

Is the composition of two closed geodesics itself a closed geodesic?

Is composition of geodesics a commutative operation?

Can all geodesics be decomposed into compositions of primitive closed geodesics?

If anyone has comments or references, I would appreciate them.

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Selberg is not (yet?) in Abel's league, so we still write his last name capitalized :) –  Mariano Suárez-Alvarez Sep 25 '10 at 19:50
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@Mariano: Dear Mariano, I'm not sure about your first claim, but I agree with the second: Selberg should be capitalized. –  Matt E Sep 26 '10 at 0:46
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Dear Simon, Think about composing two great circles on a sphere. Is this typically another great circle? –  Matt E Sep 26 '10 at 0:47
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@Matt, heh: I really meant not to underestimate Selberg's value! The league I had in mind was the Uncapitalized League :) –  Mariano Suárez-Alvarez Sep 26 '10 at 11:04
    
@Mariano Quoting Selberg: "I want to make it clear that I never have read in detail Abel’s so-called Paris Memoir, which for a long time disappeared before it was recovered and published long after Abel’s death. But it is that little note, upon which the results of the Paris Memoir rest, which is so extremely elementary. There really is no comparison in the mathematical literature, I think. Such a fundamental and far-reaching theorem proved by so simple and elementary methods—it is pure magic. I cannot imagine anything that quite compares to this." –  user7980 Jul 30 '11 at 20:42

2 Answers 2

http://arxiv.org/abs/math/0407288 you may start your voyage here with 'introduction to the Selberg zeta function' it explain its role for Riemann Hypothesis and a generalization of the Poisson summation formula

http://matwbn.icm.edu.pl/ksiazki/aa/aa91/aa9132.pdf HERE it explain the Zeta function of Selberg in term of the determinant of a certain Laplacian over the surface , also the zeta regularization for determinant is used

for zeta regularization and functional determinants see http://arxiv.org/PS_cache/arxiv/pdf/1108/1108.5659v1.pdf

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Hyperbolic geodesic can be run once, twice, three times, every time resulting in a new hyperbolic geodesic formally ..., which admit a multiple of the origninal length.

The primitive one are those, which are not a multiple of any other geodesic.

This can be phrased in terms of generators of the fundamental group, which gives the translation to Fuchsian groups, i.e. $\pi_1(\Gamma \backslash \mathbb{H}) \cong \Gamma$.

Your Riemann surface has no singularieties, iff the generators of $\Gamma$ are in one to one correspondance to primitive geodesics.

So your first question: ... if and only if they are a multiple of the same primitive geodesic. But it is not possible to combine two arbitrary closed geodesics.

2nd question: ... Yes, the topological operation commutes, if it is well defined.

3rd question: ... Yes, every closed geodesic is a multiple of an unique primitive geodesic.

Reference: Iwaniec - Spectral theory of automorphic forms.

Google Fuchsian groups.

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