Prime Harmonic Modulation and the Riemann Zeta Function: Quantum States on the Critical Line

We investigate the mathematical structure of the Prime HarmonicModulation (PHM) wavefunction and establish that its amplitudecoefficients A_p = 1/sqrt(p) are the Dirichlet series coefficientsof the Riemann zeta function evaluated on the critical line Re(s) = 1/2.We prove the PHM wavefunction's normalizability boundary is exactly thecritical line, state the PHM-GUE conjecture, and provide numericalevidence from N=500 and N=2000 prime spectra. IBM Quantum hardwareexperiments validate PHM circuit properties. Patent PCT/US25/39370.