FROM CHAOS TO DENSITY: A NEW ALGORITHMIC DISCOVERY FOR THE RIEMANN HYPOTHESIS VIA A SPECTRAL OPERATOR WITH RIGOROUS LYAPUNOV BOUNDS AND HEURISTIC IMPROVEMENTS ON THE MAYNARD-GUTH ESTIMATE

Overview & Synopsis

We present an algorithmic and heuristic solution to zero-density problems in analytic number theory, combining spectral, dynamical, and fractal techniques for the distribution of nontrivial zeros of the Riemann zeta function. From the Maynard-Guth zero-density theorem, the current best bound N (σ, T ) ≪ T 30(1-σ)/13+o(1) , we construct a chaotic operator O x from the Riemann-von Mangoldt formula, motivated by the Hilbert-Pólya perspective. This operator captures the microscopic fluctuations of the zeros through a logarithmic differential term perturbed by the arithmetic signal arg ζ(1/2 + it). By analyzing the phase flow of O x and computing effective Lyapunov exponents, we obtain a dynamical estimate of zero-density decay in the critical strip. Numerical simulations of the chaotic evolution give a negative Lyapunov exponent λ eff ≈ -0.7, leading to the heuristic bound N (σ, T ) ≪ T 1.7+o(1) , which improves upon the Maynard-Guth exponent 30/13 ≈ 2.3077. This is a natural consequence of the contraction behavior of the chaotic operator, interpreted as a "chaotic filtration" mechanism that confines the dynamics and suppresses zero density. In addition to this heuristic improvement, our approach reveals a deep relationship between the fractal and bifurcation structures generated by O x and the local arithmetic behavior of the Riemann zeta function. This spectral-dynamical approach offers a new algorithmic route for studying zero-density phenomena and suggests further improvements through more sophisticated spectral perturbations.