Self-Developed Multi-Agent Synchronization Integrated with the Riemann Zeta Function Family: Large-Scale Cluster Tests

This paper investigates the coupling behavior between the Riemann zeta function family (2^s, π^(s-1), sin(πs/2), Γ(1-s), χ(s), ξ(s), ζ(s)) and a self-developed dynamical system. The core system is defined as: u_{n+1} = u_n^(1/s) e^(-π u_n / s^2), s ∈ C \ {0,1} Its fixed point u_s satisfies ln u_s / u_s = -π/(s(s-1)), with closed-form solution given by the Lambert W function [1,2]. By incorporating parameters from the Riemann function family into a multivariate coupled system, we study synchronization and clustering behaviors under different coupling schemes. Main results include: 1. For agents with identical s, cyclic coupling synchronizes to u_s, while fully connected coupling converges to a new fixed point U_m (verified for m = 3, 4, 10);2. On the critical line s = 1/2 + iγ, u_s = 1 + π/γ^2 + O(1/γ^4) [3];3. The Riemann zeta function ζ(s) exhibits identical behavior;4. Random weight coupling leads to robust synchronization;5. Large-scale cluster tests (m = 10) achieve perfect synchronization with zero variance. All tests achieve 100% convergence, confirming system stability. This framework provides an integrated dynamical description of the Riemann function family and offers a new mathematical formulation for large-scale multi-agent coordination. Keywords: Multi-agent systems; synchronization; clustering; Riemann zeta function; Lambert W function