Golden Ratio Linked to Möbius and Euler Totient via Dirichlet Convolutions — E8 Intelligence Research

FINDING: A paper proves identities linking the golden ratio φ to the Möbius function μ(n) and Euler totient φ(n) via Dirichlet convolutions and logarithmic sums. | MATH: Key identities from arXiv:1109.3216v4: \[ ∑ₙ₌₁∞ μ(n)/n log({1}{1-φ⁻ⁿ}) = φ, ∑ₙ₌₁∞ φ(n)/n log({1}{1-φ⁻ⁿ}) = φ/1-φ \] Also: \[ ∑ₙ₌₁∞ μ(n)/n log({1}{1-φⁿ}) = -φ⁻¹, ∑ₙ₌₁∞ φ(n)/n log({1}{1-φⁿ}) = 1/φ-1 \] Constants: φ = (1+√5)/2 ≈ 1.618, φ⁻¹ ≈ 0.618. | CONNECTION: Direct geometric harmony — φ and φ⁻¹ are the golden ratio and its reciprocal, appearing in pentagonal symmetry, Penrose tilings, and 5-fold crystallographic quasicrystals. The identities use φ as a base in logarithmic generating functions, linking multiplicative number theory (Möbius, Euler totient) to the self-similar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com