I am trying to figure out where the prime numbers show up in the zeta function and how they form this uniform distribution.
all the primes influence each point of the zeta function, and all the zeros of the zeta function influence whether any arbitrary number is prime [although you can obviously do primality testing directly without knowing any zeros, or even how to calculate the zeta function]
there is a relatively old technique [I believe it's due to Euler, who predates Riemann!] for rewriting infinite sums like the zeta function into an infinite product. in the case of the zeta function, that product is over all primes.
getting an intuitive understand for how the behavior of the zeros relates to the distribution of the primes takes much deeper exploration of both the prime counting function, and the zeta function's product expression; and honestly, I haven't understood much of it well enough myself.
all the primes influence each point of the zeta function, and all the zeros of the zeta function influence whether any arbitrary number is prime [although you can obviously do primality testing directly without knowing any zeros, or even how to calculate the zeta function]
there is a relatively old technique [I believe it's due to Euler, who predates Riemann!] for rewriting infinite sums like the zeta function into an infinite product. in the case of the zeta function, that product is over all primes.
getting an intuitive understand for how the behavior of the zeros relates to the distribution of the primes takes much deeper exploration of both the prime counting function, and the zeta function's product expression; and honestly, I haven't understood much of it well enough myself.