Usually Riemann hypothesis is introduced along this lines.
(1.1) Geometric progressions were known since forever
(1.2) Euler factorization links a product of primes and a sum of natural numbers
(1.3) Harmonic series diverges, thus there are infinity many primes
(1.4) Riemann expanded zeta function to whole complex plane and conjured that all non-trivial zeroes have $\Re=1/2$
So far so good. Nice and steady. How did he come to this conclusion isn't obvious, but this is why RH is an open problem. Suddenly...
(2.1) Non-trivial zeroes of zeta function impose constraints on distribution of prime numbers including the accuracy of prime counting function distribution of (a) gaps between primes, (b) twin primes ...
Indeed, (1.2) linked all natural numbers to primes. But why distribution of zeroes is relevant? Why not Fourier transform of zeta function, not distribution of ones?
In other words, I understand why a certain property of zeta function is relevant to primes. But why this certain property happen to be the distribution of zeroes?
Edit: I've read this What is so interesting about the zeroes of the Riemann $\zeta$ function? but roughly around this point
"This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their 'expected' positions."
"Roughly speaking, the explicit formula says the Fourier transform of the zeros of the zeta function is the set of prime powers plus some elementary factors."
it gets hard to follow.