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 $g(u)=1-{\left(\frac{\sin (\ddot{I}€u)}{\ddot{I}€u}\right)}^2+\hat{I}´(u)$. The recurrence $x_{n+1}=1-{\left(\frac{\sin (\ddot{I}€/x_n)}{\ddot{I}€/x_n}\right)}^2+\frac{1}{x_n}$ 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 $f(x)=1-\ddot{I}{€}^2{x}^2$ yield Gaussian Lyapunov function $V(x)=C_1{e}^{-\ddot{I}{€}^2{x}^3/3}$ with LaSalle invariance bounding zeros in $[0,1]$; large $x$ exhibit exponential growth $\hat{I}{»}_n\to \ln (\ddot{I}{€}^2/6)$. Entropy analysis confirms GUE level repulsion with zero entropy for small initial conditions. Comparative validation against actual ${\hat{I}}_n^3$ achieves errors $<10^{-100}$, while spectral density $\ddot{I}(E)\sim \frac{\log E}{2\ddot{I}€}$ 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 $\hat{I}\P (1/2+it)$.

v2: English improved. New quantum operator section added. Abstract/Future Research enhanced + references updated. Thanks to Pedro Cáceres for collaboration and arXiv admin for the opportunity to improve our work