Liouville Function Partial Sums: A Sufficient Condition for Riemann Hypothesis — E8 Intelligence Research

FINDING: The Riemann hypothesis remains unsolved; the only substantive new mathematical lead is an arithmetical approach via the Liouville function's partial sums, while popular sources (videos) offer no novel proof. | MATH: Sufficient condition for RH: if the partial sum \( L(x) = ∑n ≤ x λ(n) \) satisfies \( L(x) = O(x1/2 + ε) \) for all \( ε > 0 \), then all nontrivial zeros of \( ζ(s) \) lie on \( (s) = 1/2 \). The paper (arXiv:0906.4155v7) also derives a formula "curiously related" to this condition — likely involving \( ∑n ≤ x λ(n)/n^s \) or a Mellin transform of \( L(x) \). No explicit new constants or ratios appear in the search results. | CONNECTION: The critical line \( (s) = 1/2 \) is the geometric midpoint of the critical strip \( 0 < (s) < 1 \) — a perfect 0.500 symmetry, not the golden ratios. However, the zeta function's zeros are conjectured to form a 1D quasi-lattice on that line, with average spacing \( 2π/ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com