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: \[ \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \ln\left( \frac{1}{1 - \varphi^{-n}} \right) = \varphi \] and its reciprocal counterpart: \[ \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \ln\left( \frac{1}{1 - \varphi^{n}} \right) = -\varphi^{-1} \] where \(\varphi = \frac{1+\sqrt{5}}{2} \approx 1.6180339887\). Additionally, the Euler totient function appears in a related form: \[ \sum_{n=1}^{\infty} \frac{\varphi(n)}{n} \ln\left( \frac{1}{1 - \varphi^{-n}} \right) = \frac{\varphi}{\varphi-1} = \varphi^2 \approx 2.6180339887 \] 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