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Prime-Weighted Interference Patterns Inspired by the Euler Product

Jouni J. Takalo

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.

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 controls amplitude growth, slope scaling, and stability of crossings. A heuristic asymptotic argument identifies as a distinguished balance regime separating high-energy and over-damped behavior. The results concern the defined model itself.
Paper Structure (10 sections, 3 theorems, 11 equations, 2 figures)

This paper contains 10 sections, 3 theorems, 11 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model Definitions
  3. Mechanism of Destructive Interference
  4. Amplitude and Balance Regimes
  5. Slope Scaling
  6. Numerical Experiments
  7. Phase alignment illustration
  8. Weight exponent comparison
  9. Discussion
  10. Conclusion

Key Result

Proposition 1

If for some $t_0$ a large subset of primes satisfies then $S_{P,x}(t)$ develops a deep destructive-interference well near $t_0$, potentially producing a zero-like crossing.

Figures (2)

  • Figure 1: Progressive superposition of prime components in the phase-referenced signal $W_{N,1/2}(t)$. (a) Partial sums obtained by including primes incrementally from $p=2$ to $p=97$. Individual cosine components attain their minima at distinct phase locations determined by their frequencies $\log p$. Although these minima do not coincide exactly, clusters of near-minima generate pronounced destructive-interference wells. (b) The same process viewed under the phase reference $\theta(t)$, illustrating how increasing the prime cutoff sharpens the interference minimum and may produce a zero-like crossing.
  • Figure 2: Weight-exponent dependence of the raw prime cosine signal $S_{P,x}(t)=\sum_{p\le P} p^{-x}\cos(t\log p)$ on $t\in[140,160]$. Left: $P=100$ (25 primes). Right: $P=10^6$ (78,498 primes). Red dashed lines mark known zeta-zero ordinates (external reference). For $x<\tfrac{1}{2}$, higher primes retain substantial weight and the large-ensemble signal becomes high-energy and rapidly oscillatory. For $x=\tfrac{1}{2}$, destructive-interference minima sharpen with increasing $P$ while overall amplitude growth remains controlled. For $x>\tfrac{1}{2}$, higher prime contributions are strongly attenuated, producing a smoother and effectively over-damped signal.

Theorems & Definitions (8)

  • Definition 1: Raw prime cosine signal
  • Definition 2: Phase-referenced signal
  • Remark 1
  • Definition 3: Zero-like crossing
  • Proposition 1: Heuristic zero-like crossing mechanism
  • Proposition 2: Balance exponent
  • Proposition 3: RMS slope scaling (heuristic)
  • Remark 2