Number Theory

    

The Mirror Wave Function of Prime Numbers

Authors: Ahmed Souissi

Abstract. While exploring Dirichlet L-functions as part of a final-year project, I stumbledupon a surprising property: a wave function ψp(x) = χ(p)eiγx, where χis a non-trivial Dirichlet character modulo q, γ is a non-trivial zero of L(s,χ), and p is a prime, produces a discrete Fourier transform ψp(k) with a dominant peak at k ≡ p−1 mod q. This "mirror symmetry"suggests a deep arithmetic structure linking primes to their modular inverses. I formalize this observation with a quantitative conjecture, provide numerical evidence for q = 5,13,17, andoffer a partial theoretical analysis using Gauss and Kloosterman sums. Potential applicationsin quantum physics and cryptography are discussed.

Comments: 3 Pages.

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Submission history

[v1] 2025-06-16 01:24:22

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