Asymptotic Stability of the Zeta Function via Differential Perturbations of Step Functions

This paper presents a heuristic framework for examining the stability conditions of the Riemann Zeta function ζ(s) within the critical strip. Addressing the non-convergence of the classical Dirichlet series for Re(s) ≤ 1, we employ a double regularization technique: (1) Abel summation damping (e−εn) to ensure absolute convergence, and (2) a telescopic identity to rigorously incorporate the unitary constant term into the summation. By decomposing the oscillatory error term induced by the discrete nature of the integers into Polylogarithmic functions (Lis), we transform the arithmetic problem into a spectral stability analysis. We analyze the Laplacian of the regularized perturbation energy, deriving trigonometric expansions which suggest that the “energy” of differential perturbations remains bounded if and only if the system satisfies a spectral balance condition at Re(s) = 1/2.