Quasicrystal Scattering and the Riemann Zeta Function

We construct a one-dimensional quasicrystal by placing scatterers at positions $\chi_n = \ln(p_n)$, the logarithms of the primes. This map compresses the primes to approximately constant density and yields a Fourier transform that is directly parameterized by the Riemann zeta function: the scattering amplitude $\hat{\chi}_L(k) = \sum p_n^{-2\pi ik}$, and the non-trivial zeros of $\zeta(s)$ enter as poles of $-\zeta'/\zeta$ in the spectral decomposition, producing peaks at positions $\gamma/2\pi$. We evaluate this Fourier transform analytically in the limit $L\to\infty$ via Perron's formula and the residue theorem, showing that the normalized amplitude assigns each non-trivial zero $\rho_m$ a coefficient proportional to $p_L^{\beta_m - 1/2}$. We then prove, using the unconditional Fourier self-duality identity $\mathcal{F}[\mathcal{F}[\chi]] = \chi(-\,\cdot\,)$ in the space of tempered distributions, that these coefficients must all be $O(1)$, which forces $\beta_m = 1/2$ for every non-trivial zero.