What does this research mean for the field?

The Riemann Hypothesis is false because there exist non-trivial zeros of the Riemann zeta function that do not lie on the critical line. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.CHALLENGES_CONSENSUS.

What question did this study set out to answer?

To evaluate the validity of the Riemann Hypothesis by analyzing contour integrals involving the logarithmic derivative of the Riemann xi function within the critical strip.

August 23, 2026Open Access

The Riemann hypothesis is false

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Authors

MRManuel Hernández RosalesUniversidad Nacional Autónoma de México

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Implication

Theoretical analysis demonstrates the existence of non-trivial zeros off the critical line, refuting the Riemann hypothesis.

Key Points

To evaluate the validity of the Riemann Hypothesis by analyzing contour integrals involving the logarithmic derivative of the Riemann xi function within the critical strip.
Analyzed a rectangular contour integral of (π / sin(πs))(ξ'(s + 1/2) / ξ(s + 1/2)) within the critical strip using Cauchy's Residue Theorem and Parseval's equality.
Applied Littlewood's growth estimates for the xi function to assess the limiting behavior of the horizontal integral components as height approaches infinity.
Demonstrated that the vertical boundary integrals yield a non-vanishing series with singular behavior as ε approaches 0, whereas horizontal boundary components vanish in the infinite height limit.
Showed that the resulting non-zero total contour integral contradicts Cauchy's Residue Theorem under the assumption of the hypothesis, establishing the existence of non-trivial zeros off the critical line.

Cite This Study

Manuel Hernández Rosales (2026) studied this question.

synapsesocial.com/papers/6a8aae407677a341144471b9 https://doi.org/10.5281/zenodo.22048726

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The Riemann hypothesis is false

Discussion

Member takes

Implication

Key Points

Cite This Study

The Riemann hypothesis is false

Discussion

Member takes

Implication

Key Points

Cite This Study

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