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: \[ \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \log \left( \frac{\phi^n - (-\phi)^{-n}}{\sqrt{5}} \right) = \log \phi \] \[ \sum_{n=1}^{\infty} \frac{\varphi(n)}{n} \log \left( \frac{\phi^n - (-\phi)^{-n}}{\sqrt{5}} \right) = \log \phi \] where \(\mu(n)\) is Möbius function, \(\varphi(n)\) is Euler totient, \(\phi = (1+\sqrt{5})/2 \approx 1.618\), and the inner term is the Binet form for Fibonacci numbers \(F_n\). Also: \[ \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \log \left( \frac{1}{1 - \phi^{-n}} \right) = \log \phi \] \[ \sum_{n=1}^{\infty} \frac{\varphi(n)}{n} \log \left( \frac{1}{1 - \phi^{-n}} \right) = \log \phi \] Constants: \(\phi = 1.6180339\ldots\), \(\phi^{-1} = 0.6180339\ldots\), \(\sqrt{5} \approx 2.23607\). | CONNECTION: Direct geometric harmony link via golden ratio \(\phi\) and its reciprocal \(0.618\), appearing i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com