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

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) ≪ T30(1−σ)/13+o(1), we construct a chaotic operator Ox 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 Ox 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 λeff ≈ −0.7, leading to the heuristic bound N(σ, T) ≪ T1.7+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 Ox 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.