Zeta-Regularized Determinants for the Gamma Factor and Prime Modes A Relative Trace Program for the Riemann ξ Funct

We propose a spectral-determinant program motivated by the Riemann hypothesis (RH) that combines (i) an arithmetic model in which primes are treated as independent bosonic modes and (ii) a geometric “container” model responsible for the completed (archimedean) factor G (s): = π^ (−s/2) Γ (s/2). On the arithmetic side we consider the oneparticle space ℓ² (P) and the diagonal operator Hˆₚ |p⟩ = (ln p) |p⟩. Its second quantization yields a bosonic Fock-space Hamiltonian whose partition function equals ζ (s) for Re (s) > 1. On the geometric side we show, using classical Hurwitz–zeta identities, that G (s) admits an explicit representation as a ζ-regularized determinant, and we identify a solvable realization via the symmetric trigonometric P¨oschl–Teller inverse-square operator on (0, 1) (with parameter g > 3/2). Finally we introduce an analytic regularization parameter m and a kernel f (n, m) = nᵐ/m − sin (πm) / (πm²), and we formulate a conjectural operator-theoretic kernel/trace regularization aimed at combining the archimedean determinant factor and the prime-mode partition function into a single relative determinant object. In a rigorous formulation based on ζ-regularized determinants and the classical analytic continuation of ζ (s), we define a relative determinant Ξdet (s) that coincides with the completed zeta function ξ (s) on Re (s) > 1 and hence extends uniquely to an entire function.