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August 16, 2026Open Access

Three-Dimensional Numerical Phase Geometry System: Theory of Geometric Stability for Non-Trivial Zeros — Applications and Limitations in the Study of the Riemann Hypothesis 三维数相位几何体系:非平凡零点分布的几何稳定性理论及其在黎曼假设研究中的应用与局限

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Authors

ZLZhongqiang Liu

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Implication

Computational study demonstrates geometric stability of Riemann zeta non-trivial zeros along the critical line, suggesting a phase-space framework for the Riemann Hypothesis.

Key Points

  • Investigate the distribution and stability of non-trivial zeros of the Riemann zeta function using a Three-Dimensional Numerical Phase Geometry framework.
  • Mapped Riemann zeta function behavior within the critical band into a three-dimensional phase space across twenty systematic numerical experiments (Experiments A to T).
  • Evaluated timelike and spacelike geometric metrics, isotropic boundaries, contour integrals, Lipschitz propagation interval arithmetic, and Gaussian Unitary Ensemble (GUE) quantum chaos statistics.
  • Demonstrated that the potential function V(σ) achieves a global minimum at σ=0.5 with a positive restoring force (F>0) observed for 100% of tested non-trivial zeros.
  • Showed zero timelike points at the critical line σ=0.5 versus a 0.798 spacelike proportion at σ=1.0, with the zero-spacing distribution of 491 zeros matching GUE statistics and N(T) matching at 99.8%.
  • Verified strict ζ-convexity for the first 100 non-trivial zeros within σ=0.5±0.15 via interval arithmetic, while detailing four specific theoretical gaps between numerical verification and analytic proof.

Cite This Study

Zhongqiang Liu (2026) studied this question.

synapsesocial.com/papers/6a817a60f2fb91fc834ae2dehttps://doi.org/10.5281/zenodo.21932397
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