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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 10th 2014
    • (edited Dec 10th 2014)

    Added pointers to Selberg zeta function for the fact that, under suitable conditions over a 3-manifold, the exponentiated eta function exp(iπη D(0))\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?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 11th 2014

    John Baez kindly points out that the analogy between the Selberg zeta and the Artin L given in the nnLab here had been highlighed much in

    • Darin Brown, Lifting properties of prime geodesics, Rocky Mountain J. Math. Volume 39, Number 2 (2009), 437-454 (euclid)

    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.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 11th 2014

    Perhaps you could explain the notation n Γ(g)n_\Gamma(g)?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 11th 2014

    Right, sorry, I still need to add definition of this and a few other terms.

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