Quasicrystal Scattering and the Riemann Zeta Function
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911711468978176 |
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| author | Shaughnessy, Michael |
| author_facet | Shaughnessy, Michael |
| contents | We construct a one-dimensional quasicrystal by placing scatterers at positions $χ_n = \ln(p_n)$, 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 $\hatχ_L(k) = \sum p_n^{-2πik}$, and the non-trivial zeros of $ζ(s)$ enter as poles of $-ζ'/ζ$ in the spectral decomposition, producing peaks at positions $γ/2π$. 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 $ρ_m$ a coefficient proportional to $p_L^{β_m - 1/2}$. We then prove, using the unconditional Fourier self-duality identity $\mathcal{F}[\mathcal{F}[χ]] = χ(-\,\cdot\,)$ in the space of tempered distributions, that these coefficients must all be $O(1)$, which forces $β_m = 1/2$ for every non-trivial zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03673 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quasicrystal Scattering and the Riemann Zeta Function Shaughnessy, Michael Quantum Physics Materials Science High Energy Physics - Theory Mathematical Physics We construct a one-dimensional quasicrystal by placing scatterers at positions $χ_n = \ln(p_n)$, 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 $\hatχ_L(k) = \sum p_n^{-2πik}$, and the non-trivial zeros of $ζ(s)$ enter as poles of $-ζ'/ζ$ in the spectral decomposition, producing peaks at positions $γ/2π$. 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 $ρ_m$ a coefficient proportional to $p_L^{β_m - 1/2}$. We then prove, using the unconditional Fourier self-duality identity $\mathcal{F}[\mathcal{F}[χ]] = χ(-\,\cdot\,)$ in the space of tempered distributions, that these coefficients must all be $O(1)$, which forces $β_m = 1/2$ for every non-trivial zero. |
| title | Quasicrystal Scattering and the Riemann Zeta Function |
| topic | Quantum Physics Materials Science High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2410.03673 |