2023/11/09 by Zeraoulia Rafik, Rafik, Zeraoulia, Caceres, Pedro
Mathematics · #11M26-37D45-37N20-37G15-37M20 #Advanced Mathematical Identities #Analytic Number Theory Research #Chaotic Dynamics (nlin.CD) #FOS: Mathematics #FOS: Physical sciences #General Mathematics (math.GM) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2406.12852
openalex publication_date 2023/11/09 · openalex created_date 2024/06/22 · openalex updated_date 2026/07/28
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 - ( (sin(πu))/(πu) )2 + δ(u). The recurrence xn+1 = 1 - ( (sin(π/xn))/(π/xn) )2 + (1)/(xn) 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 - π2 x2 yield Gaussian Lyapunov function V(x) = C1 e-π2 x3/3 with LaSalle invariance bounding zeros in [0,1]; large x exhibit exponential growth λn → ln(π2/6). Entropy analysis confirms GUE level repulsion with zero entropy for small initial conditions. Comparative validation against actual γn achieves errors <10-100, while spectral density ρ(E) ∼ (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).