What question did this study set out to answer?

The aim is to provide a geometric proof of the Riemann Hypothesis by analyzing the trajectories of the Riemann ξ-function.

April 10, 2026Open Access

Geometric Proof of the Riemann Hypothesis: Angular Singularity and Spiral Divergence from the Critical Base 1/2, 0

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Authors

YKYilWook Kim

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Implication

This proof demonstrates the behaviour of Riemann zeros using geometric analysis of trajectories in the complex plane, suggesting new insights into their nature.

Key Points

The aim is to provide a geometric proof of the Riemann Hypothesis by analyzing the trajectories of the Riemann ξ-function.
Defined a radial coordinate system mapping the real and imaginary axes.
Analyzed the trajectory of the functional I(s) = ξ(s)² from the critical base point s0 = [1/2,0].
Identified a Critical Capture Angle θc to assess phase-velocity and divergence.
Investigated phase-capture dynamics and the Geometric Escape Threshold.
Proved the existence of a unique Phase-Locked Path at 90° in the I-plane.
Identified a fundamental bifurcation at θc ≈ 47.96°, affecting path behavior.
Showed that trajectories with θ > θc exhibit a capture into a stable orbit near the origin, while those with θ < θc diverge directly.

Cite This Study

YilWook Kim (2026) studied this question.

synapsesocial.com/papers/69d8948f6c1944d70ce05750 https://doi.org/10.5281/zenodo.19446787

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Implication

This proof demonstrates the behaviour of Riemann zeros using geometric analysis of trajectories in the complex plane, suggesting new insights into their nature.

Geometric Proof of the Riemann Hypothesis: Angular Singularity and Spiral Divergence from the Critical Base 1/2, 0

Discussion

Member takes

Implication

Key Points

Cite This Study

Geometric Proof of the Riemann Hypothesis: Angular Singularity and Spiral Divergence from the Critical Base 1/2, 0

Discussion

Member takes

Implication

Key Points

Cite This Study

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