What does this research mean for the field?

The unconditional lower bound for the proportion of nontrivial zeros of the Riemann zeta function that are simple and lie on the critical line is 67.25126275289546%, as formally verified using the Lean 4 theorem prover. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To establish and formally verify an improved unconditional lower bound for the proportion of nontrivial zeros of the Riemann zeta function that are simple and lie on the critical line.

September 2, 2026Open Access

A Lean-Verified Refinement of the Unconditional Simple-Zero Bound for the Riemann Zeta Function (67.251%)

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To establish and formally verify an improved unconditional lower bound for the proportion of nontrivial zeros of the Riemann zeta function that are simple and lie on the critical line.
Retained a nonnegative spectral remainder in the rank-trace framework and lower-bounded it using local Gram interactions between consecutive simple zeros on the critical line.
Computed the bound via path-forest aggregation coupled with an exact rational gap program.
Formally kernel-checked the inequality in Lean 4 using two independent paths: a five-box analytic proof and a fixed-point reflection proof across 170 accepted leaves.
Established an unconditional, Lean-verified lower bound of 67.25126275289546% for the proportion of nontrivial zeros that are simple and situated on the critical line.
Improved upon the prior Alpöge-Furman Montgomery-Taylor-window benchmark of 67.2500703679% by approximately 0.001192 percentage points.

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Implication

Formal mathematical verification demonstrates a 67.251% simple-zero bound on the critical line for the Riemann zeta function, indicating tighter unconditional limits on nontrivial zeros.

Cite This Study

Takayuki Komada (2026) studied this question.

synapsesocial.com/papers/6a97e305c562ede874ec78c0 https://doi.org/10.5281/zenodo.22209568

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Implication

Formal mathematical verification demonstrates a 67.251% simple-zero bound on the critical line for the Riemann zeta function, indicating tighter unconditional limits on nontrivial zeros.

A Lean-Verified Refinement of the Unconditional Simple-Zero Bound for the Riemann Zeta Function (67.251%)

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Discussion

Implication

Cite This Study

A Lean-Verified Refinement of the Unconditional Simple-Zero Bound for the Riemann Zeta Function (67.251%)

Key Points

Discussion

Implication

Cite This Study

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