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

TL;DR

This work formulates a Montgomery-inspired dynamical system $x_{n+1}=1-ig(rac{\sin(\pi/x_n)}{\pi/x_n}\big)^2+\frac{1}{x_n}$ to model nontrivial zeros of the Riemann zeta function and their GUE-style repulsion. Through bifurcation analysis, Lyapunov exponents, and entropy studies, the authors reveal a coexistence of stable limit cycles near $x=0$ and chaotic trajectories for small initial conditions, with a Gaussian-like Lyapunov structure $V(x)=C_1 e^{-\pi^2 x^3/3}$ bounding zeros to $[0,1]$ and exponential growth for large $x$ with $\lambda=\ln(\pi^2/6)$. The model achieves ultra-high accuracy against actual zeros ($<10^{-100}$ errors) and reproduces the asymptotic spectral density $\rho(E)\sim \frac{\log E}{2\pi}$, supporting the conjectured quantum-operator perspective on zeta zeros and their pair correlations. The paper also discusses three avenues to realize a Pólya–Hilbert-type operator from the Montgomery dynamics—discretization to tridiagonal operators, a semiclassical $H=xp+V(x)$ construction, and neural-operator learning—laying groundwork for potential numerical tests of the Riemann hypothesis via spectral properties.

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

Figures (13)

• Figure 1: Approximate solutions for $x_0 = 0.00000005$
• Figure 2: Approximate solutions for $x_0 = 0.00000005$
• Figure 3: Lyapunov Exponents for Case 1 (x = 0.5)
• Figure 4: Lyapunov Exponents for Case 2 ($x = 0.0000005$)
• Figure 5: Analytical Solutions and Error for Case $x=0.5$

Theorems & Definitions (1)

• Theorem 1