Selberg and Twisted Ruelle Zeta Functions: Meromorphic Continuation on Hyperbolic Surfaces — E8 Intelligence Research

Abstract

FINDING: Selberg zeta function generalizes Riemann zeta to hyperbolic surfaces; twisted Ruelle zeta at zero yields meromorphic continuation for compact hyperbolic surfaces of genus ≥2. MATH: Selberg zeta function \( Z_S(s) = \prod_{\{\gamma\}} \prod_{k=0}^\infty (1 - e^{-(s+k)\ell(\gamma)}) \), where \(\gamma\) are primitive closed geodesics on a compact hyperbolic surface, \(\ell(\gamma)\) their lengths. Twisted Ruelle zeta \( R_\chi(s) \) with representation \(\chi\) of \(\pi_1(X)\); meromorphic continuation to \(\mathbb{C}\) proven. Genus \(g \geq 2\) implies constant negative curvature \(-1\). CONNECTION: Hyperbolic surfaces have constant curvature \(-1\), linking to hyperbolic geometry. No direct golden ratio or base-60 constants appear. However, geodesic length spectra often involve arithmetic Fuchsian groups, which can exhibit ratios like \(\sqrt{2}, \sqrt{3}, \phi\) in special cases (e.g., quaternion algebras). The Selberg trace formula connects lengths to eigenvalues of La Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com