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

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 $x_{n+1}=1-{\left(\frac{\sin (\pi \mathrm{/}{x}_n)}{\pi \mathrm{/}{x}_n}\right)}^2+\frac{1}{x_n}$ emulates eigenvalue repulsion as a quantum operator analogue realizing the Pólya-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 $V(x)=C_1{e}^{-{\pi }^2{x}^3\mathrm{/}3}$ with LaSalle invariance bounding zeros in $[0,1]$; large $x$ exhibit exponential growth ${\lambda }_n\to \ln ({\pi }^2\mathrm{/}6)$. 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 $\zeta (1\mathrm{/}2+it)$.