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analytic function analytic proof assume Bekenstein bound black-hole boundaries that confine circumscribes the system complete Schwinger functions confine the field converges d-dimensional Euclidean space dimensions Dirichlet boundary condition divergent contribution E(mo eigenvalue Epstein zeta-function Euclidean field theory example the Ref finite free energy function 7d functional integral functional of complete Gaussian functional given by Eq heat-kernel high temperature holographic principle hypercube independent-value generating functional Klauder's result law of thermodynamics let us define low temperature behavior macroscopic boundaries maximum value meromorphic function modified Epstein-Hurwitz zeta parameter presence of macroscopic Qo(cr quantum bound quantum field renormalized zero-point energy satisfies a quantum satisfies the inequality scalar field theory self-interacting scalar field situations a quantum spacetime specific entropy satisfies spectral zeta function spectral zeta-function sphere that circumscribes strong-coupling expansion strong-coupling perturbative expansion strong-coupling regime Substituting Eq summation temperature limit term theory per unit third law valid value generating functional Xavier Sigaud