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given a surface with metric $ g_{ab} $ i would like to evaluate the functional determinant of the Laplacian in the form

$ - \partial _{s} \zeta (0,E^{2})=\log\det( \Delta + E^{2}) $

then i need to evaluate the Theta function $ \Theta (t)= \sum_{n} \exp(-tE_{n})$ for $t >0$ real

since this is too hard :( my idea is to use the approximation

$ \Theta (t) = \frac{1}{(2\pi)^{d}} \int dpdx\exp(-tH(x,p)) $

with the Hamiltonian of the surface $ H(x,p)= \sum_{a,b}g^{a,b}(x)p_{a}p_{b} $ summation over indices $a$ and $b$ is assumed :)

my question is if this approximation for the 'Theta function' or Heat Kernel of the Laplacian of the surface is valid, i need the Theta function in order to evaluate the determinant of the Laplacian by zeta-regularization.

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