Riemann Hypothesis: No New Proof, Only Conditional Liouville Sum Criterion — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: No new proof or approach to the Riemann hypothesis emerges from these sources; they are primarily expository videos and one arXiv paper (0906.4155v7) offering a conditional sufficient condition via the Liouville function partial sums. | MATH: The arXiv paper's core is a sufficient condition: RH holds if the partial sum L(x) = ∑n ≤ x λ(n) satisfies L(x) = O(x1/2+ϵ) for all ϵ > 0 — equivalent to the known equivalence between RH and the Mertens function bound (though Mertens conjecture is false, this Liouville form is weaker and still open). The paper also presents a "curiously related" formula, but no explicit new constant or ratio is derived. | CONNECTION: The zeta function's nontrivial zeros lie on the critical line ℜ(s) = 1/2. The value 1/2 is not a golden-ratio harmonic, but the critical line's symmetry s ↔ 1−s is a reflection symmetry. The Liouville function \( \lambda(n) = (-1)^{\Omega(n)} \
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026riemann,
title = {{Riemann Hypothesis: No New Proof, Only Conditional Liouville Sum Criterion — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23587181},
url = {https://zenodo.org/records/23587181},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Riemann Hypothesis: No New Proof, Only Conditional Liouville Sum Criterion — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23587181
UR - https://zenodo.org/records/23587181
AB - FINDING: No new proof or approach to the Riemann hypothesis emerges from these sources; they are primarily expository videos and one arXiv paper (0906.4155v7) offering a conditional sufficient condition via the Liouville function partial sums. | MATH: The arXiv paper's core is a sufficient condition: RH holds if the partial sum L(x) = ∑_n ≤ x λ(n) satisfies L(x) = O(x^1/2+ϵ) for all ϵ > 0 — equivalent to the known equivalence between RH and the Mertens function bound (though Mertens conjecture is false, this Liouville form is weaker and still open). The paper also presents a "curiously related" formula, but no explicit new constant or ratio is derived. | CONNECTION: The zeta function's nontrivial zeros lie on the critical line ℜ(s) = 1/2. The value 1/2 is not a golden-ratio harmonic, but the critical line's symmetry s ↔ 1−s is a reflection symmetry. The Liouville function \( \lambda(n) = (-1)^{\Omega(n)} \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -