Spectral determinant and spectral Zeta functions on hyperbolic surfaces and polygons

I will give a short introduction into spectral zeta functions for manifolds and domains and explain the relation between heat kernel bounds, finite propagation speed bounds, and bounds for the spectral zeta function. In principle these results can be used to compute the spectral determinant rather accurately with rigorous error estimates. I will discuss briefly some questions about extremal properties of spectral zeta functions.