Aliasing and Entanglement in the Riemann Zeta Function: A Signal Processing Framework for Zero Formation

This paper presents a mathematical framework that analyzes the Riemann zeta function through principles of signal processing, focusing on aliasing effects that emerge in the summation of the Dirichlet series for its real and imaginary parts. These aliasing effects, formalized as the alias zeta function, preserve the phase structure and amplitude of the original Dirichlet series along the critical line, mirroring the original summands with precise phase alignment. This framework reveals a new axis of symmetry in the zeta function, where aliasing effects and original summands interact in a structured, mirrored relationship. The periodic rotation of these aliasing components induces a distinct periodicity in the zeta function, leading to regular interference patterns between the real and imaginary parts and their corresponding aliasing components. A central insight of this framework is that the real and imaginary parts of the zeta function are not independent but are intrinsically coupled through the rotational dynamics of their aliasing effects. This coupling enforces simultaneous cancellations along the critical line, where the precise phase alignment between original and aliasing components results in synchronized zero formation. Furthermore, the framework provides an explanation for the absence of zeros off the critical line. Off the critical line, the essential proportional relationship between the real and imaginary parts and their aliasing components is disrupted, breaking the symmetry required for simultaneous cancellation. This disruption prevents the synchronized alignment needed for zero formation, confining non-trivial zeros to the critical line.