# Determinant of the Laplacian of a surface is this correct?

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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