Yes, high-level descriptions of the RH are often vague about the connection between the zeta function and the prime numbers, but the short version is that, for every point on on the complex plane, there is an associated real-valued "Riemann harmonic function"; if you sum up the Riemann harmonic functions of the non-trivial zeros of the zeta function, you get the exact prime-counting step function, and this function is much more predictable if the RH is true.
Cool, thank you for confirming what I knew!!! It's nice to have the connection between intuition and formal :)
It is strange that there's such a tight analogy between the zeroes (gaps*) and the primes (gaps in multiples). Suggests some patterning even more fundamental than just gaps of integer magnitude multiples.
Although maybe it's just another description of the same thing for now and can't pierce through to yield more insight...Unsure! I suspect there's something more fundamental out there waiting to be discovered that will illuminate the whole thing :)
* i guess you can call zeroes gaps of a function in a few ways: simply they are just a gap in the magnitude, as they zero it out; or, in the sense that they are 'gaps in the curvature of whatever it integrates to / is the differential of'; but also I suppose a nicer analogy is that the zeroes are 'factors' of the function [Y(x) = 0 at x = z, then (x-z) is a factor of Y right?] maybe that's the tighter connection, but would RH then imply some close connection between factors of complex functions or polynomials and prime factors?? Anyway, just explorin I totally understand if you don't wish to reply no probs! :)
