An Operator–Fractal Algorithm for Heuristic Refinement of the Maynard–Guth Zero-Density Bound

We propose a new algorithmic method for tackling zero-density problems in analytic number theory by integrating spectral and dynamical approaches to the distribution of zeros of the Riemann zeta function. We extend the Maynard--Guth zero-density result, which currently provides the best known bound \[ N(\sigma,T)\ll T^{30(1-\sigma)/13+o(1)}, \] by constructing a chaotic operator $O_x$ based on the Riemann--von Mangoldt formula and Hilbert--Pólya reasoning. This operator captures the microscopic fluctuations of the zeros via a logarithmic differential term \[ d\log\zeta\Big(\frac12+it\Big) \] perturbed by \[ \arg \zeta\Big(\frac12+it\Big). \]Our approach exploits the phase evolution of $O_x$ to determine \emph{effective Lyapunov exponents}, which dynamically indicate the rate of zero-density decay within the critical strip. Simulating the chaotic flow of $O_x$, we obtain a heuristic zero-density bound \[ N(\sigma,T)\ll T^{1.7+o(1)}, \] which improves upon the Maynard--Guth exponent $30/13 \approx 2.3077$. This improvement arises from the contraction behavior of the chaotic operator and its negative Lyapunov exponent, reflecting the confining dynamics and local repulsion of the nontrivial zeros.Beyond refining the heuristic bound, this method unveils a profound connection between fractal and chaotic structures associated with the operator and the arithmetic behavior of Riemann zeta zeros.