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Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces. (English) Zbl 0629.58029

The author gives an estimate for the Selberg zeta function Z(R,s) of a compact Riemann surface of genus \(g\geq 2\) valid for s in a neighborhood, depending only on the genus, of \(s=1\). The author uses a formula of McKean relating Z’/Z to the heat kernel K of R and determines then the behaviour of K when R tends to infinity in the moduli space \(M_ g\) of stable curves. From this he deduces the behaviour of the spectral measure, and the functional determinant, as R tends to infinity in \(M_ g\). Using this and the expression of the holomorphic parameter for the transverse direction to the compactification divisor of \(M_ g\) in terms of Fenchel-Nielsen coordinates (that he etablishes), the author proves that the bosonic Polyakov integral \((d=26)\) is infinite. Similar results are obtained by Hejhal who uses the B-groups as model for the degeneration of Riemann surfaces [D. A. Hejhal, Preprint, Chalmers Univ. Göteborg (1987)].
Reviewer: M.Burger

MSC:

58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
81T08 Constructive quantum field theory
11M35 Hurwitz and Lerch zeta functions
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