A Fractal-Holographic Representation of the Riemann Zeta Function via a Self-Similar Operator

This article introduces an alternative representation of the Riemann zeta function \(\zeta(s)\) based on a fractal-holographic operator \(\mathcal{F}_s\). This operator couples the Möbius function to a logarithmic self-similar phase generated by a weighted iterated function system (IFS). The construction reformulates the classical Dirichlet series as the fixed point of a compact operator acting on a Banach space of holomorphic functions. The Hausdorff dimension of the associated auxiliary attractor converges to \(1/2\) on the critical line, providing a topological sufficient condition for the localization of the non-trivial zeros. We prove strict equivalence with the classical definition in the half-plane \(\operatorname{Re}(s) > 1\), elucidate the mechanism of meromorphic analytic continuation, and present high-precision numerical validations. This formalism opens a new pathway for the spectral analysis of the zeros by identifying them as the eigenvalues of a fractal attractor of critical dimension.