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

This paper introduces a mathematical framework for analyzing the Riemann zeta function using principles of signal processing, focusing on aliasing effects that arise during the summation of the Dirichlet series for the real and imaginary parts. These observed aliasing effects, mathematically described by the alias zeta function, are shown to be identical representations of the original Dirichlet series summands, preserving their frequency and amplitude structure. This discovery suggests a new axis of symmetry in the zeta function, where the aliasing effects and the original summands interact in a perfectly mirrored fashion. The periodic rotation of the aliasing effects induces a previously unnoticed periodicity in the behavior of the zeta function, leading to regular interference between the real and imaginary parts and their respective aliasing counterparts. This framework also explains the simultaneous cancellation of the real and imaginary parts as a result of the symmetric alignment of their aliasing effects. A key finding of this framework is that the real and imaginary parts of the zeta function are not independent, but are entangled through the rotational dynamics of their aliasing effects. The periodic rotation of the aliasing effects creates a synchronized interaction between the real and imaginary parts, where their oscillatory behavior is governed by the alignment and interplay of both original and aliasing components. This entanglement leads to structured interference patterns, where the real and imaginary parts cancel each other out precisely when their aliasing effects are perfectly aligned.