det(−Δ) via ζ_Δ(s) = Σλ_n^{−s} — analytic continuation to s=0
ζ_A(s) = Σ_{λ_n>0} λ_n^{-s}
log det A = −ζ'_A(0)
det(−Δ) = —
The spectral zeta function encodes all eigenvalues. Its value at s=0 gives the Weyl asymptotic counting, and −ζ'(0) = log det A is used in quantum field theory (1-loop determinants, Casimir effect).
Spectral Zeta & Ray-Singer Determinant (1971): For an elliptic operator A with eigenvalues {λ_n}, the spectral zeta function ζ_A(s) = Σλ_n^{-s} converges for Re(s) large and admits meromorphic continuation. The functional determinant det A = exp(−ζ'_A(0)) regularizes the formal infinite product Πλ_n. For the Laplacian on a Riemannian manifold, ζ(0) is a topological invariant (via heat kernel expansion). The Casimir energy of the EM field is computed as (1/2)ζ_{∂²}(−1/2). On tori and spheres, spectral zeta values relate to classical number-theoretic zeta functions (Riemann, Hurwitz, Epstein).