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Added pointers to Selberg zeta function for the fact that, under suitable conditions over a 3-manifold, the exponentiated eta function $\exp(i \pi \, \eta_D(0))$ equals the Selberg zeta function of odd type.
Together with the fact at eta invariant – For manifolds with boundaries this says that the Selberg zeta function of odd type constitutes something like an Atiyah-style TQFT which assigns determinant lines to surfaces and Selberg zeta functions to 3-manifolds.
This brings me back to that notorious issue of whether to think of arithmetic curves as “really” being 2-dimensional or “really” being 3-dimensional: what is actually more like a Dedekind zeta function: the Selberg zeta functions of even type or those of odd type?
John Baez kindly points out that the analogy between the Selberg zeta and the Artin L given in the $n$Lab here had been highlighed much in
Page 9 there has a table with all the key ingredients.
Except maybe for one detail: there the analogy is made between number fields and hyperbolic surfaces. Whereas I think now it works a bit better still for hyperbolic 3-manifolds.
Perhaps you could explain the notation $n_\Gamma(g)$?
Right, sorry, I still need to add definition of this and a few other terms.
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