What does this research mean for the field?

More than 67.345129% of the nontrivial zeros of the Riemann zeta function are simple and lie on the critical line, and more than 83.672564% are distinct. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Establish unconditional asymptotic lower bounds for the proportion of nontrivial zeros of the Riemann zeta function that are simple and on the critical line, as well as distinct zeros overall.

August 31, 2026Open Access

An Unconditional 67.345129% Lower Bound for Simple Zeros of the Riemann Zeta Function on the Critical Line

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Establish unconditional asymptotic lower bounds for the proportion of nontrivial zeros of the Riemann zeta function that are simple and on the critical line, as well as distinct zeros overall.
Applied a concave finite-dimensional Gram-defect profile separately inside consecutive seven-point windows prior to block aggregation within the Alpöge–Furman finite-compression framework.
Resolved a proof-critical six-variable inequality with block length m = 367 and local target 0.0079109 using fail-closed branch-and-bound algorithms with outward-rounded interval arithmetic.
Verified all 64,954 coarse cells and 129,909 midpoint locations across six interval-table streams using an independent closed-form checker and exact rational arithmetic.
Proved an unconditional lower bound showing that greater than 67.345129% (exact rational constant 2820183620839/4187657898650) of nontrivial zeros are simple and lie on the critical line.
Established that more than 83.672564% (exact rational constant 7007841519489/8375315797300) of nontrivial zeros are distinct when measured against total zero count with multiplicity.

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Implication

Computer-assisted analysis demonstrates an unconditional 67.34% lower bound for simple zeros in the Riemann zeta function, indicating that most zeros on the critical line are non-multiple.

Cite This Study

Ketan Singh (2026) studied this question.

synapsesocial.com/papers/6a95427bf20e493292a74cae https://doi.org/10.5281/zenodo.22119371

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Implication

Computer-assisted analysis demonstrates an unconditional 67.34% lower bound for simple zeros in the Riemann zeta function, indicating that most zeros on the critical line are non-multiple.

An Unconditional 67.345129% Lower Bound for Simple Zeros of the Riemann Zeta Function on the Critical Line

Key Points

Discussion

Implication

Cite This Study

An Unconditional 67.345129% Lower Bound for Simple Zeros of the Riemann Zeta Function on the Critical Line

Key Points

Discussion

Implication

Cite This Study

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