Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces
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 [\textit{D. A. Hejhal}, Preprint, Chalmers Univ. Göteborg (1987)].
scientific article; zbMATH DE number 4024359
scientific article