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Super-zeta functions and regularized determinants associated with cofinite Fuchsian groups with finite-dimensional unitary representations. (English) Zbl 1473.58018

Let \(S\) be a hyperbolic surface of finite volume and let \(\Delta\) be its Laplace-operator. For \(s\in\mathbb C\) with \(\operatorname{Re}(s)>1\), I. Efrat showed in [Commun. Math. Phys. 119, No. 3, 443–451 (1988; Zbl 0661.10038)], that the product \(\prod_\sigma((\sigma(1-\sigma)-s(1-s))\) can be regularized, where \(\sigma\) runs over the set of all eigenvalues of \(\Delta\) and all scattering resonances. He called this product \(\det(\Delta-s(1-s))\) and showed that it essentially equals the Selberg zeta function.
In the present paper it is shown that the products over the eigenvalues alone can also be regularized and the same holds for the resonances. This is done in a rather explicit form, ending up in a concise form of determinant expression.

MSC:

58J52 Determinants and determinant bundles, analytic torsion
11F72 Spectral theory; trace formulas (e.g., that of Selberg)

Citations:

Zbl 0661.10038
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References:

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