Prime-Weighted Interference Patterns Inspired by the Euler Product

Abstract

We study a prime-weighted oscillatory model inspired by structural aspects of the Euler product of the Riemann zeta function. The model defines finite superpositions of prime-frequency modes and exhibits zero-like crossings produced by destructive interference. We analyze how the weight exponent $x$ controls amplitude growth, slope scaling, and stability of crossings. A heuristic asymptotic argument identifies $x=\tfrac12$ as a distinguished balance regime separating high-energy and over-damped behavior. The results concern the defined model itself.