Selberg Zeta Universality Links Prime Geodesics to Number-Theoretic Primes — E8 Intelligence Research

Abstract

FINDING: Selberg zeta function universality for arithmetic groups mirrors Riemann zeta universality, linking prime geodesic lengths to number-theoretic primes via spectral geometry. MATH: - Selberg zeta function: \( Z_S(s) = \prod_{\{\gamma\}} \prod_{k=0}^\infty \left(1 - e^{-(s+k)\ell(\gamma)}\right) \), where \(\ell(\gamma)\) are lengths of primitive closed geodesics on a hyperbolic surface. - Universality theorem (Drungilas-Garunkstis-Kacenas, 2013): For the modular group \(\mathrm{SL}(2,\mathbb{Z})\), \(Z_S(s)\) approximates any non-vanishing analytic function on a compact set within the critical strip, analogous to Voronin's theorem for \(\zeta(s)\). - Geodesic length spectrum: \(\ell(\gamma) = 2\cosh^{-1}(n/2)\) for \(n \in \mathbb{Z}^+\) in the modular group case, yielding lengths like \(\log((n+\sqrt{n^2-4})/2)\). - Golden ratio appears via hyperbolic geometry: The shortest closed geodesic length on the modular surface is \(2\arccosh(3/2) \approx 2.6339\), but the rati Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com