Abstract
This essay examines whether the spectral and renormalization-group structures in the thermal backreaction of a scalar field in de Sitter spacetime, as studied by Kalogirou, provide a viable bridge to the Riemann Hypothesis through the dynamical zeta function framework.
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The Dynamical Zeta Function Analogy
This essay explores a speculative bridge between cosmological backreaction in de Sitter (dS) space and the Riemann Hypothesis (RH), mining the source paper for patterns relevant to dynamical systems and transfer operators. Kalogirou's work (arXiv:2601.15878v1) studies a semi-classically backreacted dS metric where scalar field perturbations satisfy a differential equation of Whittaker type. In the holographic regime, the paper constructs flow equations for the dual quantum field theory that match the beta functions of the three-dimensional Sp(N) vector model.
The proposed analogy connects the Whittaker spectral problem to the spectral theory of transfer operators appearing in Ruelle's dynamical zeta functions, and relates the holographic RG flow to the deformation theory of dynamical systems. The hope is that the "frozen attractor" regime and the recurrence relations for mode coefficients might mirror the spectral determinants that control zeta zero locations.
Analogy Strength: Suggestive Metaphor (weak). The correspondence fails to achieve formal analogy because the de Sitter isometry group SO(1,4) does not act discretely to produce an Euler product, and the Sp(N) RG flow lacks the arithmetic structure associated with the Riemann zeta function's critical line. The essay provides a precise analysis of these obstructions.
This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.