3 · Add the waves yourself

The prime staircase, and the explicit formula's rebuild from the first zero-waves.

Try this. Add waves one at a time. With none, the rebuild is a featureless ramp; each wave adds one frequency, and by fifty the corners of the staircase have assembled themselves out of pure tones.

The gray curve steps up at every prime and prime power, weighted the standard way; the blue curve is the smooth ramp minus one cosine per zero. The identity behind the convergence is Riemann's explicit formula, and the tests check the miss really shrinks.