The zeta determinant of the Dirichlet-to-Neumann map of a surface with boundary equals the determinant of the discrete part of its boundary Hilbert transform, a product of period ratios on the double surface.
BelishevThe Calderon problem for two-dimensional manifolds by the BC-method
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The determinant of the Dirichlet-to-Neumann map for a surface with boundary and periods of holomorphic differentials on its double
The zeta determinant of the Dirichlet-to-Neumann map of a surface with boundary equals the determinant of the discrete part of its boundary Hilbert transform, a product of period ratios on the double surface.