The Mirror Wave Function of Prime Numbers

Abstract

Ahmed Souissi*

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.

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