The Hilbert-Polya conjecture traditionally seeks a self-adjoint operator whose L2 - spectrum corresponds to the non-trivial zeros of the Riemann zeta function ζ(s). However, on non-compact hyperbolic manifolds with cusps, waves leak into the continuous spectrum, rendering a pure discrete L2-spectrum physically and mathematically unviable. Returning to the acoustic paradigm of Riemann and Helmholtz, we propose a shift from bound states to scattering resonances. We construct a network of hyperbolic Helmholtz resonators—tubular neighborhoods around primitive closed geodesics associated with prime numbers. We demonstrate that the critical line Re(s) = 1/2 emerges naturally as a null-metric (isotropic) surface where the geometric degeneration of the resolvent metric enforces the unitarity of the scattering matrix. In this paper, we provide the rigorous mathematical foundations for this framework, proving the four core conditions (C1–C4) of the Artemov Conjecture: the geometric involution, the existence of the null-metric surface, the self-adjointness of the perturbed Laplacian via Kato’s monotone convergence theorem, and the algebraic uniqueness of the critical line.
Oleg V. Artemov (Sun,) studied this question.