What does this research mean for the field?

Unconditionally, asymptotically at least 67.33% of the nontrivial zeros of the Riemann zeta function are simple zeros on the critical line, and at least 83.66% of the nontrivial zeros are distinct. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To establish unconditional lower bounds on the asymptotic proportions of simple zeros located on the critical line and distinct zeros of the Riemann zeta function.

September 5, 2026Open Access

Two-Certificate Lower Bounds for Simple and Distinct Zeros of the Riemann Zeta Function

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To establish unconditional lower bounds on the asymptotic proportions of simple zeros located on the critical line and distinct zeros of the Riemann zeta function.
Applied a finite Hermitian compression combined with a rank–trace argument to separate simple zeros on the critical line from off-line symmetric pairs.
Constructed rigorous local inequalities via two finite point configurations matched with a bandwidth-eight spectral trace–energy envelope.
Certified finite numerical inequalities using rigorous interval computations without assuming the truth of the Riemann hypothesis.
Proved unconditionally that asymptotically at least 67.3320366508509707% of the nontrivial zeros are simple zeros lying on the critical line.
Proved unconditionally that asymptotically at least 83.6660183254254853% of the nontrivial zeros are distinct.

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Implication

Analytical proof demonstrates higher proportions of simple and distinct zeros in the Riemann zeta function, indicating major progress on prime number distribution.

Cite This Study

Dirk Schäfer (2026) studied this question.

synapsesocial.com/papers/6a9bd49d6b95aff0620ec6fb https://doi.org/10.5281/zenodo.22280746

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Implication

Analytical proof demonstrates higher proportions of simple and distinct zeros in the Riemann zeta function, indicating major progress on prime number distribution.

Two-Certificate Lower Bounds for Simple and Distinct Zeros of the Riemann Zeta Function

Key Points

Discussion

Implication

Cite This Study

Two-Certificate Lower Bounds for Simple and Distinct Zeros of the Riemann Zeta Function

Key Points

Discussion

Implication

Cite This Study

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