Journal of Electrical Electronics Engineering(JEEE)
ISSN: 2834-4928 | DOI: 10.33140/JEEE
Impact Factor: 1.2
Research Article - (2025) Volume 4, Issue 4
The Mirror Wave Function of Prime Numbers
Received Date: Jun 05, 2025 / Accepted Date: Jul 07, 2025 / Published Date: Jul 17, 2025
Copyright: ©©2025 Ahmed Souissi. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation: Souissi, A. (2025). The Mirror Wave Function of Prime Numbers. J Electrical Electron Eng, 4(4), 01-03.
Abstract
This paper explores a wave function ψ p (x) = χ(p)e iγx , where χ is a non-trivial Dirichlet character modulo q, γ is a non-trivial zero of the L-function L(s,χ), and p is a prime coprime to q. The discrete Fourier transform ψ ˜ p (k) exhibits a dominant peak at k ≡ p −1 (mod q), suggesting a mirror symmetry in the arithmetic of primes. Motivated by community feedback, we correct earlier errors in sum evaluations, provide detailed numerical evidence using LMFDB data, and refine speculative applications in physics and cryptography. All calculations are rigorously verified, and code is available upon request.
Introduction
During a final-year project on Dirichlet L-functions, we observed an intriguing property: a wave function defined for prime numbers produces a Fourier transform with a peak at their modular inverses. This paper formalizes this “mirror symmetry” as a conjecture, inspired by analogies between L-function zeros and quantum spectra [3]. Community feedback identified inaccuracies in the original analysis, prompting this revised version. We correct the mathematical framework, provide comprehensive numerical evidence, and temper speculative applications to ensure rigor suitable for further study.
Dirichlet L-functions generalize the Riemann zeta function and encode prime distribution modulo q [1]. Their non-trivial zeros, conjectured to lie on 
, are pivotal in analytic number theory [2]. Drawing on quantum analogies [3], we define a wave function ψp (x) and analyze its Fourier transform. This work corrects errors in asymptotic behavior and sum evaluations, details numerical methods, and aims to contribute to discussions on prime arithmetic.