Given a compact surface of constant negative curvature M we denote by Δ: L2(M)→L2(M) the associated Laplacian. The operator -Δ has discrete eigenvalues 0=λ-<λ1≤λ2≤...→+∞to which one can associate a (functional) determinant, as follows. The Dirichlet series ηM(s)=Σ∞n=1λ-sn is known to converge providing Re(s) is sufficiently large, and possesses a meromorphic extension to the entire complex plane C[6]. Using this extension we define the determinant (of the Laplacian) by det (ΔM)=exp (-dη/ds(0)).
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