Prime-Weighted Interference Patterns Inspired by the Euler Product

TL;DR

This paper introduces a prime-weighted oscillatory model inspired by the Euler product, analyzing interference patterns and stability controlled by a weight exponent, with a focus on the critical balance at x=1/2.

Contribution

It presents a novel prime-weighted interference model inspired by the Euler product, analyzing its behavior and stability across different regimes.

Findings

The model exhibits zero-like crossings due to destructive interference.

The weight exponent x controls amplitude growth and crossing stability.

x=1/2 is identified as a critical balance point separating different behaviors.

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.