A Deterministic Matrix Operator Incorporating Prime Number Data: Construction and Spectral Statistics
Fuente:
Zenodo
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Recurso digital |
| Lingua: | inglese |
| Pubblicazione: |
Zenodo
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866901667513892864 |
|---|---|
| author | Karazoupis, Miltiadis |
| author_facet | Karazoupis, Miltiadis |
| contents | <p><span>The Hilbert-Pólya conjecture proposes that the non-trivial zeros of the Riemann zeta function lie on the critical line because their imaginary parts correspond to the eigenvalues of a Hermitian operator. Furthermore, numerical evidence and the Montgomery-Odlyzko law suggest these eigenvalues exhibit statistical properties matching the Gaussian Unitary Ensemble (GUE) of Random Matrix Theory (RMT). Motivated by the lack of an explicit construction for such an operator, we introduce a novel, finite-dimensional, deterministic matrix model designed to incorporate information from prime numbers while respecting fundamental symmetries. Specifically, we construct a 2N x 2N Hermitian matrix H with inherent particle-hole symmetry, built from a N x N complex matrix A whose elements are functions of the logarithms of the first N primes. We numerically diagonalize H for N up to 1200 and analyze the nearest-neighbor spacing distribution P(s) of its positive eigenvalues. Our results confirm the operator's Hermiticity and spectral symmetry. The eigenvalue spacings exhibit clear level repulsion (P(s)→0 as s→0), a key feature of RMT and quantum chaos. However, Kolmogorov-Smirnov tests show statistically significant deviations from the GUE Wigner distribution (p-values < 1e-5 for N=100, decreasing to < 1e-94 for N=1200). Moreover, we observe no convergence towards GUE statistics as N increases within the tested range. This suggests that while the model captures essential structural features and generates non-trivial spectral correlations, this specific deterministic construction based on prime data does not fall into the GUE universality class, highlighting the profound challenge in constructing an operator that fully replicates the conjectured spectral properties of the Riemann zeros.</span></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_15098112 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Deterministic Matrix Operator Incorporating Prime Number Data: Construction and Spectral Statistics Karazoupis, Miltiadis Riemann Hypothesis Hilbert-Pólya Conjecture Random Matrix Theory GUE Spectral Statistics Prime numbers Matrix Models Computational Number Theory <p><span>The Hilbert-Pólya conjecture proposes that the non-trivial zeros of the Riemann zeta function lie on the critical line because their imaginary parts correspond to the eigenvalues of a Hermitian operator. Furthermore, numerical evidence and the Montgomery-Odlyzko law suggest these eigenvalues exhibit statistical properties matching the Gaussian Unitary Ensemble (GUE) of Random Matrix Theory (RMT). Motivated by the lack of an explicit construction for such an operator, we introduce a novel, finite-dimensional, deterministic matrix model designed to incorporate information from prime numbers while respecting fundamental symmetries. Specifically, we construct a 2N x 2N Hermitian matrix H with inherent particle-hole symmetry, built from a N x N complex matrix A whose elements are functions of the logarithms of the first N primes. We numerically diagonalize H for N up to 1200 and analyze the nearest-neighbor spacing distribution P(s) of its positive eigenvalues. Our results confirm the operator's Hermiticity and spectral symmetry. The eigenvalue spacings exhibit clear level repulsion (P(s)→0 as s→0), a key feature of RMT and quantum chaos. However, Kolmogorov-Smirnov tests show statistically significant deviations from the GUE Wigner distribution (p-values < 1e-5 for N=100, decreasing to < 1e-94 for N=1200). Moreover, we observe no convergence towards GUE statistics as N increases within the tested range. This suggests that while the model captures essential structural features and generates non-trivial spectral correlations, this specific deterministic construction based on prime data does not fall into the GUE universality class, highlighting the profound challenge in constructing an operator that fully replicates the conjectured spectral properties of the Riemann zeros.</span></p> |
| title | A Deterministic Matrix Operator Incorporating Prime Number Data: Construction and Spectral Statistics |
| topic | Riemann Hypothesis Hilbert-Pólya Conjecture Random Matrix Theory GUE Spectral Statistics Prime numbers Matrix Models Computational Number Theory |
| url | https://doi.org/10.5281/zenodo.15098112 |