Riemann Hypothesis Remains Unproven; Partial-Sum Approach Conditional — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: No new proof or disproof of the Riemann Hypothesis (RH) emerges; the search yields pedagogical expositions and one arithmetical partial-sum approach (Liouville function) that remains conditional and incomplete. | MATH: RH states all non-trivial zeros of ζ(s) satisfy Re(s)=1/2. The arithmetical paper (arXiv:0906.4155v7) gives a *sufficient* condition: if the partial sums of the Liouville function λ(n) satisfy Σ_{n≤x} λ(n) = O(x^{1/2+ε}) for all ε>0, then RH holds. The paper also derives a formula "curiously related" to that condition, but no proof of the bound is given. No new constants, ratios, or exact equations beyond the standard ζ(s) = Σ n^{-s} (Re(s)>1) and its functional equation ζ(s)=2^s π^{s-1} sin(πs/2) Γ(1−s) ζ(1−s). | CONNECTION: The critical line Re(s)=1/2 is a *geometric bisector* of the critical strip 0
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
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BibTeX
@misc{caldin2026riemann,
title = {{Riemann Hypothesis Remains Unproven; Partial-Sum Approach Conditional — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635605},
url = {https://zenodo.org/records/23635605},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Riemann Hypothesis Remains Unproven; Partial-Sum Approach Conditional — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635605
UR - https://zenodo.org/records/23635605
AB - FINDING: No new proof or disproof of the Riemann Hypothesis (RH) emerges; the search yields pedagogical expositions and one arithmetical partial-sum approach (Liouville function) that remains conditional and incomplete. | MATH: RH states all non-trivial zeros of ζ(s) satisfy Re(s)=1/2. The arithmetical paper (arXiv:0906.4155v7) gives a *sufficient* condition: if the partial sums of the Liouville function λ(n) satisfy Σ_{n≤x} λ(n) = O(x^{1/2+ε}) for all ε>0, then RH holds. The paper also derives a formula "curiously related" to that condition, but no proof of the bound is given. No new constants, ratios, or exact equations beyond the standard ζ(s) = Σ n^{-s} (Re(s)>1) and its functional equation ζ(s)=2^s π^{s-1} sin(πs/2) Γ(1−s) ζ(1−s). | CONNECTION: The critical line Re(s)=1/2 is a *geometric bisector* of the critical strip 0 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -