Riemann Hypothesis: No New Proof, Only Conditional Liouville Sum Criterion — E8 Intelligence Research

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 \( L(x) = \sum_{n \le x} \lambda(n) \) satisfies \( L(x) = O(x^{1/2+\epsilon}) \) for all \( \epsilon > 0 \) — 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 \( \Re(s) = 1/2 \). The value \( 1/2 \) is not a golden-ratio harmonic, but the critical line's symmetry \( s \leftrightarrow 1-s \) 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