Number Theory

    

The Mirror Wave Function of Prime Numbers

Authors: Guiffra Patrick

We explore the application of the **Discrete Fourier Transform (DFT)** to a class of modular wave functions. For an integer q ≥ 2, a Dirichlet character χ modulo q, and an integer p coprime to q, we introduce the modular wave function ψp(x) = χ(p)exp[(i2πp_1/q)x, where p−1 is the modular inverse of p modulo q.We rigorously demonstrate that the DFT of ψp(x) is a **Kronecker delta peak** with a value of χ(p) √q, located precisely at the frequency k = p^−1 (mod q), and zero everywhere else. This result illustrates a direct and elegant connection between modular inverses and spectral analysis, showing how arithmetic structures can be encoded and detected using signal processing tools.

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[v1] 2025-08-28 20:31:46

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