Riemann Zeta Zeros & Prime Spectral Music

The non-trivial zeros of ζ(s) on the critical line Re(s)=1/2 encode the primes. Their imaginary parts are the "frequencies" of prime distribution — Explicit Formula in action.

Parameters

Explicit Formula

ψ(x) = x − Σ_ρ x^ρ/ρ − ln(2π)

ρ = 1/2 + iγₙ (RH zeros)

Each zero contributes an oscillation of wavelength ~ 2π/γₙ to the prime staircase ψ(x)

Known Zeros (γₙ)

Connection

RH: all zeros on Re=1/2 ↔ primes are as "random" as possible. The zeros are eigenvalues of an unknown Hermitian operator (Hilbert-Pólya). GUE statistics of zeros = chaotic quantum billiards.