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: \[ \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \log\left(\frac{1}{1-\phi^{-n}}\right) = \phi, \quad \sum_{n=1}^{\infty} \frac{\varphi(n)}{n} \log\left(\frac{1}{1-\phi^{-n}}\right) = \frac{\phi}{1-\phi} \] Also: \[ \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \log\left(\frac{1}{1-\phi^{n}}\right) = -\phi^{-1}, \quad \sum_{n=1}^{\infty} \frac{\varphi(n)}{n} \log\left(\frac{1}{1-\phi^{n}}\right) = \frac{1}{\phi-1} \] 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