[Project] A spectral engineering approach to the Riemann Hypothesis: I simulated a self-adjoint quantum potential up to X_max = 10^9 to recover the zeros with 10^-8 stability. Full text and dataset published on Zenodo
Hi everyone,
I wanted to share a project I’ve been independently working on for a while. As an engineer with a deep fascination for the interplay between physics and analytic number theory, I’ve always been drawn to the Hilbert-Pólya conjecture.
Instead of treating the problem through pure abstract deduction, I’ve approached it from a spectral engineering perspective. I built a parameter-free quantum confinement potential V(u) derived from the Riemann Explicit Formula, using the exact prime counting function π(x) regularized via a continuous Weierstrass-Gaussian transform.
The goal was to construct a self-adjoint Hamiltonian operator H whose discrete spectral signature maps directly onto the non-trivial zeros of the Riemann zeta function (λₙ ~ γₙ).
💻 The Simulation & Deep Grid Scaling
I’ve recently pushed the numerical script to a deep-grid optimization ceiling of X_max = 1.0 × 10⁹, using a spatial grid resolution of N = 16,384 points.
Even under these high-dimensional space restrictions and hardware limitations (constrained to a 12.6 GB RAM desktop environment), the system has shown remarkable structural stability. The localized variance (Δ = λₙ - γₙ) completely lacks asymptotic drift or localized divergence, maintaining an invariant truncation error order of Δ ~ 10⁻⁷ to 10⁻⁸ across the entire processed spectral range.
Here is a quick look at the live tracking log from the sparse linear algebra solver (scipy.sparse.linalg.eigsh) recovering the resonance peaks:
=== Z-SUSY Explicit-Formula 1e9 PUSH (500 Zeros) ===
[+] Computing resonances (500 iterations, processing...) ...
[1/500] 14.134725 → 14.134725 (-0.00000029)
[11/500] 52.970321 → 52.970321 (-0.00000002)
[21/500] 79.337375 → 79.337375 (+0.00000001)
...
[151/500] 321.160134 → 321.160134 (-0.00000021)
[311/500] 557.564659 → 557.564659 (+0.00000018)
[321/500] 572.419984 → 572.419985 (+0.00000060)
🏛️ The Theoretical Backbone
The computational success isn't isolated. I have detailed the underlying continuous operators in a formal mathematical framework divided into three key stages:
Stage 1 (Asymptotic Confinement): Proving that the continuous potential diverges positively (lim_{u → ∞} V(u) = +∞), establishing an unbreachable confinement wall that guarantees a purely discrete spectrum.
Stage 2 (Strict Self-Adjointness): Utilizing the Kato-Rellich perturbation stability theorem and strict Dirichlet boundary conditions at the origin to ensure the spectrum remains strictly real.
Stage 3 (Spectral Duality): Mapping the roots via a regularized Weierstrass-Hadamard product determinant to tie them to the completed Riemann ξ-function.
I've also addressed the common "circularity / bootstrap challenge" in the text, outlining how subsequent stages will focus on completely decoupling the direct reliance on π(x) to achieve full arithmetic independence.
📦 Open Science & Data Availability
In the spirit of complete empirical transparency, I have published the open-access manuscript alongside the core optimized Python execution script and the high-precision research dataset.
Official DOI / Publication: https://doi.org/10.5281/zenodo.20933920
I would love to hear your thoughts, criticisms, or suggestions on the functional analysis side or the grid-scaling optimization. If anyone is working on similar spectral approaches to the Riemann Hypothesis, let's connect!