Diffraction of Log-Flattened Primes: From Scattering Amplitudes to Pair Correlation

Consider the measure μ = ∑{p} w(p) δ{log p} with weights w(p) chosen to produce approximately constant intensity in log-coordinates. Develop its autocorrelation and diffraction (Fourier transform of the autocorrelation), asking whether the diffraction decomposes into absolutely continuous, singular continuous, and pure-point parts. Use explicit formulas and pair-correlation heuristics to link the absolutely continuous part to Montgomery’s pair correlation; track arithmetic progressions (mod q) and “jumping champions” as candidates for Bragg-like features; test for singular continuous (quasicrystalline) components predicted by mid-scale structure. This project builds a rigorous diffraction framework for the log-flattened prime set, informed by equidistribution phenomena and modern random-matrix heuristics. It asks not just whether ζ is visible in scattering but what the spectral type of prime diffraction is and how it encodes pair-correlation, progressions, and gap statistics. Diffraction provides a unifying observable sensitive to correlations across scales; detecting a singular continuous component would signal quasi-order beyond GUE, while a purely absolutely continuous spectrum (except discrete spikes) would argue against mid-scale quasi-periodicity. The impact is a new cross-disciplinary diagnostic for prime correlations, with clean predictions testable numerically and interpretable via number-theoretic conjectures, potentially becoming a standard observable to compare number-theoretic models on equal footing.

References:

1. Arithmetic Consequences of the GUE Conjecture for Zeta Zeros. B. Rodgers (2024). The Michigan mathematical journal.
2. On the equidistribution properties of patterns in prime numbers Jumping Champions, metaanalysis of properties as Low-Discrepancy Sequences, and some conjectures based on Ramanujan's master theorem and the zeros of Riemann's zeta function. A. Ortiz-Tapia (2023).
3. Quasicrystal Scattering and the Riemann Zeta Function. Michael Shaughnessy (2024).
4. The Connection Between Prime Distribution and Random Matrices. Shanwen Ye (2025). Theoretical and Natural Science.
5. Arithmetic Consequences of the GUE Conjecture for Zeta Zeros. B. Rodgers (2024). The Michigan mathematical journal.