Golden Ratio Identities Linking Möbius, Euler Totient, and Natural Log — E8 Intelligence Research

FINDING: Golden ratio identities linking Möbius function, Euler totient, and natural logarithm in number theory. | MATH: Key identities from arXiv:1109.3216v4 include: \[ ∑ₙ₌₁∞ μ(n)/n log ( {φ^n - (-φ)⁻ⁿ}{√5} ) = log φ \] \[ ∑ₙ₌₁∞ φ(n)/n log ( {φ^n - (-φ)⁻ⁿ}{√5} ) = log φ \] where \(μ(n)\) is Möbius function, \(φ(n)\) is Euler totient, \(φ = (1+√5)/2 ≈ 1.618\), and the inner term is the Binet form for Fibonacci numbers \(F_n\). Also: \[ ∑ₙ₌₁∞ μ(n)/n log ( {1}{1 - φ⁻ⁿ} ) = log φ \] \[ ∑ₙ₌₁∞ φ(n)/n log ( {1}{1 - φ⁻ⁿ} ) = log φ \] Constants: \(φ = 1.6180339…\), \(φ⁻¹ = 0.6180339…\), \(√5 ≈ 2.23607\). | CONNECTION: Direct geometric harmony link via golden ratio \(φ\) and its reciprocal \(0.618\), appearing i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com