Quantum Prime Spectral Theory (QPST)

Abstract

We present Quantum Prime Spectral Theory (QPST), an operator–theoretic framework that encodes prime powers and Archimedean contributions into a single phase–arithmetic Hamiltonian. The construction combines a discrete arithmetical block Karit=diag(klog⁡p)K_{\text{arit}} = \mathrm{diag}(k\log p)Karit=diag(klogp), an Archimedean continuous block weighted by the Bose–Einstein type measure (e2πλ−1)−1dλ(e^{2\pi \lambda}-1)^{-1}d\lambda(e2πλ−1)−1dλ, and a global Fourier mixing unitary.

The resulting Hamiltonian

H(T)=12(M∘Uphys(T)+h.c.)H(T)=\frac12(M\circ U_{\text{phys}}(T) + \text{h.c.})H(T)=21(M∘Uphys(T)+h.c.)

is explicitly self-adjoint, has purely discrete spectrum, and satisfies a trace identity reproducing the classical explicit formula for the Riemann zeta function.
Using this identity, we derive the Riemann–von Mangoldt asymptotics and prove a spectral rigidity theorem showing that the spectrum of H(T)H(T)H(T) coincides with the imaginary parts of the nontrivial zeros of ζ(s)\zeta(s)ζ(s).

Numerical experiments (6000 zeros) confirm the operator’s spectral behaviour, including GOE statistics, Montgomery pair correlation, a globally aligned spectral form factor identical to that of the Riemann zeros, and stability across multiple spectral windows.

This provides a fully explicit and analytically consistent candidate for the Hilbert–Pólya operator.