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Quasicrystal Scattering and the Riemann Zeta Function

Michael Shaughnessy

TL;DR

The work engineers a 1D quasicrystal-like scattering model from logarithmically shifted primes to probe the connection between prime distributions and the non-trivial zeros of the Riemann zeta function. It derives a spectral decomposition showing resonances at k ≈ γ/2π corresponding to zeros ρ=β+iγ, with peak amplitudes scaling as L^{β}. By combining the explicit formula, Perron's method, and a reconstruction framework based on the linear independence of prime characters, the authors argue that all zeros must share the same real part, which the functional equation then pinpoints at 1/2, i.e., the Riemann Hypothesis. Numerical results corroborate the predicted peak locations, and the authors provide open-access code to reproduce the findings, offering a physical perspective on a classical number-theoretic problem.

Abstract

I carry out numerical scattering calculations against a family of finite-length one-dimensional point-like arrangements of atoms, $χ(x)$, related to the distribution of prime numbers by a shift operation making the atomic density approximately constant. I show how the Riemann Zeta Function (RZF) naturally parameterizes the analytic structure of the scattering amplitude and give numerical results.

Quasicrystal Scattering and the Riemann Zeta Function

TL;DR

The work engineers a 1D quasicrystal-like scattering model from logarithmically shifted primes to probe the connection between prime distributions and the non-trivial zeros of the Riemann zeta function. It derives a spectral decomposition showing resonances at k ≈ γ/2π corresponding to zeros ρ=β+iγ, with peak amplitudes scaling as L^{β}. By combining the explicit formula, Perron's method, and a reconstruction framework based on the linear independence of prime characters, the authors argue that all zeros must share the same real part, which the functional equation then pinpoints at 1/2, i.e., the Riemann Hypothesis. Numerical results corroborate the predicted peak locations, and the authors provide open-access code to reproduce the findings, offering a physical perspective on a classical number-theoretic problem.

Abstract

I carry out numerical scattering calculations against a family of finite-length one-dimensional point-like arrangements of atoms, , related to the distribution of prime numbers by a shift operation making the atomic density approximately constant. I show how the Riemann Zeta Function (RZF) naturally parameterizes the analytic structure of the scattering amplitude and give numerical results.
Paper Structure (25 sections, 7 theorems, 29 equations, 1 figure)

This paper contains 25 sections, 7 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fourier Transform and Quasicrystals
  3. The Prime Potential
  4. The Scattering Amplitude
  5. Definition and Basic Properties
  6. Connection to the Riemann Zeta Function
  7. Contour Integration and Peak Structure
  8. Numerical Results
  9. Summary of Main Results
  10. Acknowledgements
  11. Proof of the Riemann Hypothesis
  12. Overview
  13. Preliminaries
  14. Linear Independence of Prime Characters
  15. The Reconstruction Framework
  16. ...and 10 more sections

Key Result

Proposition 2.1

The scattering amplitude decomposes as: where the sum is over non-trivial zeros $\rho$ of $\zeta(s)$, and $R_L(k)$ contains contributions from trivial zeros and is $O(L^{-1}\log L)$.

Figures (1)

  • Figure 1: Scattering amplitude $|\hat{\chi}_L(k)|^2$ showing peaks at positions corresponding to RZF zeros (red vertical lines).

Theorems & Definitions (18)

  • Definition 1.1: Quasicrystal
  • Proposition 2.1: Spectral Decomposition
  • Definition 2.2: Spectral Coefficient
  • Lemma A.5: Character Independence
  • proof
  • Corollary A.6: Full Column Rank
  • proof
  • Definition A.7: Prime Indicator Reconstruction
  • Remark A.8
  • Theorem A.9: Uniform Scaling
  • ...and 8 more