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April 16, 2026Open Access

A Pseudo-Riemannian Metric on the Critical Strip of the Riemann Zeta Function: Schwarzschild Correspondence, Geodesic Structure, and a Geometric Reformulation of the Riemann Hypothesis

Puntos clave

To define a pseudo-Riemannian metric on the critical strip of the Riemann zeta function and relate it to the Riemann Hypothesis.
Defined a pseudo-Riemannian metric tensor on the critical strip using the Riemann zeta function.
Analyzed curvature properties at key points and verified findings numerically.
Established relationships between geodesics and curvature singularities related to the Riemann Hypothesis.
The metric has a Lorentzian signature determined by the analyticity of the function.
Identified a pole at σ = 1 as a coordinate singularity analogous to Schwarzschild horizon.
Proved that the critical line Re(s) = 1/2 serves as a geodesic with variable causal characteristics.
Non-trivial zeros of ζ correspond to curvature singularities, suggesting their significance in the Riemann Hypothesis.

Resumen

We dene a pseudo-Riemannian (Lorentzian) metric tensor gij = ∂i∂jΦζ on the critical strip of the Riemann zeta function ζ(s), where Φζ (s) = − log |ζ(s)|. By the harmonicity of Φζa direct consequence of ζ being holomorphicthe metric satises gσσ + gtt = 0 identically, forcing Lorentzian signature (+, −). We establish ve main results: (1) gij has Lorentzian signature forced by analyticity; (2) the pole at σ = 1 is a coordinate singularity (curvature K nite), in exact analogy with the Schwarzschild horizon; (3) the critical line Re(s) = 1 2 is a geodesic of gij via Γ σ tt|σ=1/2 = 0, proved from the functional equation; (4) the geodesic has variable causal character with a null point at t ≈ 5.561; (5) non-trivial zeros of ζ are curvature singularities with K ∼ cn/|s − ρn| 2 . The Riemann Hypothesis is equivalent to: all curvature singularities lie on the geodesic Re(s) = 1 2 . All results are veried numerically at 25-digit precision using mpmath over a 40 × 60 grid. Keywords: Riemann zeta function, pseudo-Riemannian metric, Lorentzian signature, Schwarzschild metric, geodesic, curvature singularity, FisherRao metric, Riemann Hypothesis. MSC 2020: 11M06, 53B30, 81T20, 53C50.

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