A definition of a determinant for elliptic (typically positive) operators via the spectral zeta function: if {λ_j} are the positive eigenvalues, form ζ(s) = ∑ λ_j^{-s}, analytically continue ζ(s) to s=0 and define det' A = exp(−ζ'(0)). This regularizes the divergent infinite product of eigenvalues and produces meaningful spectral invariants.

Principle

Replace the formal product ∏ λ_j by exp(−d/ds ζ(s)|_{s=0}) where ζ(s) is the analytic continuation of the eigenvalue power-sum; the construction requires meromorphic continuation and control of small eigenvalues, and choices like spectral cut can affect phase information.

Demonstration

For the Laplacian on the circle of length L, eigenvalues scale like (2πn/L)^2; forming the zeta function, continuing to s=0 and differentiating yields the determinant depending on L in the expected way and matching known regularized products of eigenvalues.

Misapplication

Treating the zeta-regularized determinant as multiplicative under arbitrary exact sequences of operators or ignoring zero modes without modifying the definition leads to contradictions; extending the naïve formula to non-discrete spectrum or non-elliptic operators is invalid without modification.

Consequence

Zeta-regularized determinants produce finite spectral invariants used to define analytic torsion, one-loop determinants in quantum field theory, and metric-dependent scalar invariants in spectral geometry; they allow analytic continuation of otherwise divergent spectral products.

Reversal

The naive reversal is the literal infinite product ∏ λ_j which is divergent; zeta-regularization replaces this divergence by a finite, analytically meaningful value. Conversely, other regularizations (heat-kernel, dimensional) may differ by computable renormalization constants.

Boundary

Applies to operators with discrete, positive spectrum such as elliptic operators on compact manifolds (or with suitable boundary conditions); for operators with continuous spectrum or lacking meromorphic zeta continuation one must adapt the definition (scattering determinants, relative zeta functions, etc.).

Semantic Tension

Tension exists with Fredholm or trace-class determinants used in operator theory: zeta-regularized determinants regularize products of eigenvalues for elliptic operators, while Fredholm determinants apply to I + compact perturbations and have different multiplicativity and domain properties.

Synthesis

The zeta-regularized determinant converts the divergent product of eigenvalues of an elliptic operator into the finite quantity exp(−ζ'(0)) via analytic continuation of the spectral zeta function; this yields robust spectral invariants central to analytic torsion and spectral geometry.