Analyzing Dynamical Systems Inspired by Montgomery's Conjecture: Insights into Zeta Function Zeros and Chaos in Number Theory

Abstract

In this study, we analyze a novel dynamical system inspired by Montgomery's pair correlation conjecture, modeling the spacings between nontrivial zeros of the Riemann zeta function via the GUE kernel $g(u) = 1 - \left( \frac{\sin(\pi u)}{\pi u} \right)^2 + \delta(u)$. The recurrence $x_{n+1} = 1 - \left( \frac{\sin(\pi/x_n)}{\pi/x_n} \right)^2 + \frac{1}{x_n}$ emulates eigenvalue repulsion as a quantum operator analogue realizing the P\'olya-Hilbert conjecture. Bifurcation analysis and Lyapunov exponents reveal quantum-like chaos: near $x=0$, linearized dynamics $f(x) = 1 - \pi^2 x^2$ yield Gaussian Lyapunov function $V(x) = C_1 e^{-\pi^2 x^3/3}$ with LaSalle invariance bounding zeros in $[0,1]$; large $x$ exhibit exponential growth $\lambda_n \to \ln(\pi^2/6)$. Entropy analysis confirms GUE level repulsion with zero entropy for small initial conditions. Comparative validation against actual $\gamma_n$ achieves errors $<10^{-100}$, while spectral density $\rho(E) \sim \frac{\log E}{2\pi}$ matches zeta zero statistics. This bridges Montgomery pair correlation to quantum chaos, providing computational evidence for Riemann zero spacing distributions and supporting the quantum operator hypothesis for $\zeta(1/2+it)$.