Control-Theoretic Formulation of the Riemann Zeta Function: Closed-Loop Stability, Transcendent Root Loci, and Generalized Nyquist Mapping of the Critical Strip

This paper establishes a rigorous control-theoretic paradigm for the analysis of the Riemann Zeta function ζ(s). By defining the operator of Laplace as the complex frequency variable s = σ + iω, we model ζ(s) as a transcendental, infinite-dimensional open-loop plant G(s). Through the architectural implementation of a closed-loop negative proportional feedback loop, the local dynamics of the non-trivial zeros are mapped directly to system transmission zeros. We explore the temporal response under classical excitations (impulse and white noise), construct explicit transcendental Bode magnitude and phase diagrams, and outline the corresponding Root Locus behavior. Finally, we apply the Generalized Nyquist Stability Criterion to provide a unified geometric perspective on the boundary constraints of the Critical Strip (0 <σ<1).