The primes look random until you stop looking at them one at a time. Counting them — asking how many fall below a given number — produces a staircase that, smoothed, hugs a curve so clean it feels like cheating. The error between staircase and curve is where the real story lives.
That error is governed by the non-trivial zeros of the Riemann zeta function — numbers that behave less like arithmetic and more like the resonant frequencies of a drum. Number theorists borrow the language of spectra and vibration because nothing else fits as well.