Spectral Zeta-Regularized Determinant and the Riemann Zeta Function: Detailed Proof of Spectral Correspondence

This publication presents a rigorous, step-by-step proof that the spectral zeta-regularized determinant of a Dirac-type operator—engineered to encode the nontrivial zeros of the Riemann zeta function—is exactly proportional to the Riemann zeta function via an explicit entire, nonvanishing factor. The work provides a comprehensive operator-theoretic foundation unifying topology, spectral theory, and analytic number theory, and resolves the principal formal assumptions behind the spectral interpretation of the Riemann Hypothesis for a broad expert audience.