A Dirichlet Series Linking the Golden Ratio, Möbius Function, and Euler Totient — E8 Intelligence Research

FINDING: A direct analytic identity linking the golden ratio φ to the Möbius function μ(n) and Euler totient φ(n) via a Dirichlet series involving the natural logarithm. MATH: From arXiv 1109.3216v4, the core identity is: \[ ∑ₙ₌₁∞ μ(n)/n ln( {1}{1 - φ⁻ⁿ} ) = φ \] and its reciprocal counterpart: \[ ∑ₙ₌₁∞ μ(n)/n ln( {1}{1 - φⁿ} ) = -φ⁻¹ \] where \(φ = {1+√5}{2} ≈ 1.6180339887\). Additionally, the Euler totient function appears in a related form: \[ ∑ₙ₌₁∞ φ(n)/n ln( {1}{1 - φ⁻ⁿ} ) = φ/φ-1 = φ^2 ≈ 2.6180339887 \] These identities involve the natural logarithm and the generating function of partitions (the Dedekind eta function at imaginary argument related to φ). CONNECTION: - The golden ratio φ and its square φ² = 2.618... are central constan Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com