Layers of Prime Gaps and Spectral Inheritance of Noise: An Analytic–Computational Study

This paper presents an analytic and computational framework to study noise in the distribution of prime numbers through layers based on multi-step prime gaps. Given the n-th prime pn, we consider the differences g(k,n) := p(n+k) − p(n), group these distances for different values of k and normalize them by an appropriate logarithmic scale. This produces, for each k, a discrete-time signal that can be analysed with tools from signal processing and control theory (Fourier analysis, resonators, RMS sweeps, and linear combinations). The main conceptual point is that all these layers should inherit the same underlying “noise” coming from the nontrivial zeros of the Riemann zeta function, up to deterministic filtering factors depending on k. We make this idea precise in a heuristic spectral inheritance hypothesis, motivated by the explicit formula and the change of variables t = log x. We then implement the construction computationally for hundreds of thousands of primes and several layers k, and we measure the response of bandpass resonators as a function of their central frequency. The numerical results show three phenomena: (i) the magnitude spectra of the layers share prominent peaks at the same frequencies, (ii) the RMS responses of resonators for different layers display aligned “hills” in the frequency domain, and (iii) a nearly perfect flattening is obtained by taking an appropriate linear combination of layers obtained via singular value decomposition. These findings support the picture that the same spectral content is present across layers and that the differences lie mainly in deterministic gain factors. The framework is deliberately modest from the point of view of number theory: we do not attempt to prove new theorems about primes, but rather to provide a coherent signal- processing view in which standard tools from electrical engineering can be used to explore the noisy side of the prime distribution in a structured way.