Hamiltonian With Spectrum Matching Riemann Zeta Zeros — E8 Intelligence Research

Abstract

FINDING: Yakaboylu (arXiv:2309.00405) constructs an explicit Hamiltonian whose discrete spectrum is conjectured to coincide with the non-trivial zeros of the Riemann zeta function, providing a concrete operator-theoretic realization of the Hilbert–Pólya program. | MATH: The Hamiltonian is built from a self-adjoint operator acting on a Hilbert space, with eigenvalues \( E_n \) satisfying \( \zeta(1/2 + i E_n) = 0 \) (assuming the Riemann hypothesis). The construction leverages the phase-space formulation of quantum mechanics, using a potential \( V(x) \) derived from the argument of the zeta function along the critical line. Key constants: the critical line \( \Re(s) = 1/2 \), the imaginary parts \( \gamma_n \approx 14.134725, 21.022040, 25.010858, \dots \). The spectral equation is \( \hat{H} \psi_n = \gamma_n \psi_n \), with \( \hat{H} \) expressed via a Weyl quantization of a classical Hamiltonian \( H(x,p) = p^2 + V(x) \), where \( V(x) \) is implicitly defined by \( \arg \zeta(1/2 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com