Riemann Hypothesis Linked to Black Hole Entropy via Spectral Geometry and Liouville Sums — E8 Intelligence Research

Abstract

FINDING: Riemann hypothesis linked to black hole entropy via spectral geometry and a new arithmetical approach using Liouville partial sums. MATH: - Sufficient condition for RH: \( M(x) = \sum_{n \le x} \lambda(n) = O(x^{1/2+\epsilon}) \) where \(\lambda(n)\) is Liouville function. - Key formula: \( \sum_{n=1}^{\infty} \frac{\lambda(n)}{n^s} = \frac{\zeta(2s)}{\zeta(s)} \) for \(\Re(s) > 1\). - Black hole entropy: \( S = \frac{A}{4G} \) (Bekenstein-Hawking) linked to spectral density of zeros of \(\zeta(s)\) via random matrix theory (GUE). CONNECTION: - Zeros of \(\zeta(s)\) on critical line \(\Re(s)=1/2\) correspond to eigenvalues of a chaotic quantum system (GUE), mirroring energy levels of heavy nuclei and black hole quasinormal modes. - No direct geometric ratios (0.382, 0.618, etc.) or base-60 found in these sources. - Crystallographic symmetry: None explicit, but the GUE symmetry class (unitary) is a continuous symmetry group, not discrete lattice. DEPTH: 7 — Th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com