Year: 2026
Type: article
Abstract: The remarkable agreement between the local statistics of the nontrivial zeros of the Riemann zeta function and the Gaussian Unitary Ensemble (GUE) suggests the existence of a hidden arithmetic spectral system. Motivated by Ramanujan–Fourier analysis, Montgomery’s pair-correlation conjecture, Odlyzko’s numerical experiments, and the Hilbert–Pólya philosophy, we investigate a conjectural framework in which Ramanujan spectral modes provide an arithmetic basis while primelogarithmic oscillations generate spectral fluctuations analogous to those of the Riemann zeros. A family of finite-dimensional self-adjoint operators is constructed from arithmetic kernels. The resulting framework naturally connects Ramanujan sums, prime-number oscillations, random matrix theory, quantum chaos, and spectral geometry. Although the construction does not constitute a proof of the Riemann Hypothesis, it provides a concrete mathematical setting within which arithmetic spectral realizations of the zeta zeros may be explored.
Source: Applied Physics Research
Author: Sandi Setiawan
Institution: University of Cambridge
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W7224280196