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This research explores a quantum-mechanical method for generating primes and its connection to the Riemann zeros.

May 1, 20260 citationsOpen Access

The Spectral Law of Prime Numbers

Key Points

This research explores a quantum-mechanical method for generating primes and its connection to the Riemann zeros.
Introduced a quantum Hamiltonian with delta-potentials at prime positions.
Developed two algorithms: deterministic for locating prime clusters and randomized for cryptographic applications.
Measured mean generation times of 16.97 ms for 1024-bit and 284 ms for 2048-bit primes.
Both algorithms achieved 100% accuracy in prime generation.
The correlation of eigenvalues with Riemann zeros was R² > 0.9999.
Outperformed conventional prime generation methods significantly.

Abstract

A Complete Evolution from Quantum Hamiltonian to 100% Accurate Prime Generation A quantum‑mechanical Hamiltonian with delta‑potentials at prime positions is introduced as a theoretical motivation. Its eigenvalues correlate with the Riemann zeros with \ (R² > 0. 9999\) on a holdout set, providing numerical evidence consistent with the Hilbert–Pólya conjecture without claiming a proof. We present two complementary algorithms: Code 1 (deterministic) locates prime clusters via spectral levels for reproducible research; Code 2 (randomized) implements Riemann's explicit formula for cryptographic applications. Both achieve 100% accuracy with mean generation times of 16. 97 ms (1024-bit) and 284 ms (2048-bit), significantly outperforming conventional methods. Keywords: Riemann Hypothesis, Spectral Law, Prime Numbers, Hilbert-Polya, Topological Invariant, Prime Generation, Cryptography

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