[impendia]

Analytic number theorist here --

"Fundamental interpretation of primes" is a bit much, but this has been understood for a long time. The short version of the story is

- The primes are closely related to the Riemann zeta function, which is more-or-less cobbled out of them;

- The Riemann zeta function has a lot more symmetry than one might initially expect, and harmonic analysis is how you prove this;

- The (still unproved) Riemann Hypothesis is that the zeta function has still more symmetry beyond what we've been able to prove.