Source Code and Numerical Results for a Parameter-Free Spectral Test of the Riemann Hypothesis via Explicit-Formula Confinement
Year: 2026
Type: preprint
Abstract: This repository contains the official Python implementation, datasets, and high-resolution spectral telemetry associated with the paper "A Numerical Exploration of a Weighted Berry-Keating-Type Operator: Finite-Size Effects and Spectral Approximations". The project introduces a novel, parameter-free numerical framework designed to test the spectral realization of the non-trivial Riemann zeros (the Hilbert-Polya conjecture) under the continuum infinite-limit condition. Moving away from traditional phenomenological potentials that require empirical phase-tuning parameters or manual background adjustments, this architecture anchors the confinement mechanism directly into the Explicit Formula of analytic number theory. The core operator consists of a symmetric dilation Berry-Keating foundation coupled with a scale-invariant effective potential derived exclusively from exact arithmetic data: the logarithmic integral and the exact prime-counting function. The smooth component establishes the quantum vacuum energy of the phase-space, while the normalized exact prime error term injects the microscopic oscillations that carry the information of the zeta zeros. Numerical experiments executed on truncated logarithmic grids demonstrate that by feeding only raw prime sieve data up to a horizon of X = 10^7, the system naturally generates a discrete real spectrum that converges to the low-lying Riemann zeros within millesimal precision (Delta approx. -0.00095 for the third zero), entirely free of ad-hoc regularization constants. Files included: riemann_explicit_framework.py: The complete, self-contained Python script utilizing an optimized Eratosthenes segmented sieve and a sparse shift-invert Arnoldi eigensolver to compute grid resonances. riemann_explicit_test_v2.jpeg: High-resolution plot showcasing the parameter-free potential topology and the exact alignment between computed eigenvalues and known Riemann zeta targets. This repository serves as an open-source computational laboratory to facilitate collaborative scaling tests under higher high-performance computing horizons.
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7165809809
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