What does this research mean for the field?

All non-trivial zeros of the Riemann zeta function lie on the critical line Re(s)=1/2, as proven through a geometric stability framework and field-induced Riemannian metrics within a three-dimensional complex space. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.ESTABLISHES_NEW_DIRECTION.

What question did this study set out to answer?

To establish a geometric stability framework for the Riemann zeta function within a three-dimensional commutative associative algebra and prove the Riemann Hypothesis using induced-metric manifolds.

August 22, 2026Open Access

三维复数中的黎曼ζ函数⼏何稳定性框架基于诱导度规流形的归谬法证明Geometric Stability Framework for the Riemann Zeta Function in 3D Complex Numbers A Proof by Contradiction via Induced-Metric Manifolds

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ZLZhongqiang Liu

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Implication

Theoretical study demonstrates minimal surface uniqueness in three-dimensional complex algebra, indicating that all non-trivial Riemann zeta zeros lie on the critical line.

Key Points

To establish a geometric stability framework for the Riemann zeta function within a three-dimensional commutative associative algebra and prove the Riemann Hypothesis using induced-metric manifolds.
Embedded the standard Riemann xi function into a three-dimensional commutative algebra with basis {1, e, i} and a (2+1)-dimensional Minkowski background metric.
Constructed a field-induced Riemannian metric from a squared-norm functional to transform zero distributions into a minimal-surface uniqueness problem on a manifold.
Applied global isometry symmetries from the functional equation alongside convexity minimizer connectedness theorems in a proof-by-contradiction framework.
Demonstrated the existence and uniqueness of the critical manifold corresponding to real part equal to one-half as the sole global minimal extremal surface.
Showed that assuming any off-critical zero exists violates the underlying convexity and isometric symmetry of the field-induced metric.
Established that all non-trivial zeros of the Riemann zeta function must lie strictly on the critical line.

Cite This Study

Zhongqiang Liu (2026) studied this question.

synapsesocial.com/papers/6a895f62ca7ade938187e0ca https://doi.org/10.5281/zenodo.22016323

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Implication

Theoretical study demonstrates minimal surface uniqueness in three-dimensional complex algebra, indicating that all non-trivial Riemann zeta zeros lie on the critical line.

三维复数中的黎曼ζ函数⼏何稳定性框架基于诱导度规流形的归谬法证明Geometric Stability Framework for the Riemann Zeta Function in 3D Complex Numbers A Proof by Contradiction via Induced-Metric Manifolds

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Cite This Study

三维复数中的黎曼ζ函数⼏何稳定性框架基于诱导度规流形的归谬法证明Geometric Stability Framework for the Riemann Zeta Function in 3D Complex Numbers A Proof by Contradiction via Induced-Metric Manifolds

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三维复数中的黎曼ζ函数⼏何稳定性框架 基于诱导度规流形的归谬法证明Geometric Stability Framework for the Riemann Zeta Function in 3D Complex Numbers A Proof by Contradiction via Induced-Metric Manifolds

Discussion

Member takes

Implication

Key Points

Cite This Study

三维复数中的黎曼ζ函数⼏何稳定性框架 基于诱导度规流形的归谬法证明Geometric Stability Framework for the Riemann Zeta Function in 3D Complex Numbers A Proof by Contradiction via Induced-Metric Manifolds

Discussion

Member takes

Implication

Key Points

Cite This Study

Also Consider

Also Consider

三维复数中的黎曼ζ函数⼏何稳定性框架基于诱导度规流形的归谬法证明Geometric Stability Framework for the Riemann Zeta Function in 3D Complex Numbers A Proof by Contradiction via Induced-Metric Manifolds

三维复数中的黎曼ζ函数⼏何稳定性框架基于诱导度规流形的归谬法证明Geometric Stability Framework for the Riemann Zeta Function in 3D Complex Numbers A Proof by Contradiction via Induced-Metric Manifolds