Noncommutative Geometry Unifies Zeta Zeros with Prime Arithmetic — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Connes' noncommutative geometry framework seeks to realize the Riemann zeta zeros as eigenvalues of a spectral operator on an adelic space, unifying arithmetic with quantum geometry. | MATH: The critical line Re(s)=1/2; zeta zeros ρ_n satisfy ζ(ρ_n)=0. Connes' trace formula: ∑_ρ h(ρ) = h(0) + h(1) + (1/2π)∫ h(r) (Γ'(1/4+ir/2)/Γ(1/4+ir/2)) dr − (1/2π)∑_p ∑_m (log p)/(p^{m/2}) ∫ h(r) e^{-imr log p} dr — an explicit formula equating spectral sums over zeros with arithmetic sums over primes. The adelic space A_Q = ℝ × ∏_p ℤ_p (restricted product) with the idele class group as symmetry. | CONNECTION: The critical line Re(s)=1/2 is the geometric midpoint of the critical strip 0
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026noncommutative,
title = {{Noncommutative Geometry Unifies Zeta Zeros with Prime Arithmetic — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23446646},
url = {https://zenodo.org/records/23446646},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Noncommutative Geometry Unifies Zeta Zeros with Prime Arithmetic — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23446646
UR - https://zenodo.org/records/23446646
AB - FINDING: Connes' noncommutative geometry framework seeks to realize the Riemann zeta zeros as eigenvalues of a spectral operator on an adelic space, unifying arithmetic with quantum geometry. | MATH: The critical line Re(s)=1/2; zeta zeros ρ_n satisfy ζ(ρ_n)=0. Connes' trace formula: ∑_ρ h(ρ) = h(0) + h(1) + (1/2π)∫ h(r) (Γ'(1/4+ir/2)/Γ(1/4+ir/2)) dr − (1/2π)∑_p ∑_m (log p)/(p^{m/2}) ∫ h(r) e^{-imr log p} dr — an explicit formula equating spectral sums over zeros with arithmetic sums over primes. The adelic space A_Q = ℝ × ∏_p ℤ_p (restricted product) with the idele class group as symmetry. | CONNECTION: The critical line Re(s)=1/2 is the geometric midpoint of the critical strip 0 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -