Do the positions of specific Riemann zeroes correspond to the positions of specific primes?
So I was watching a video about the Riemann hypothesis, and I was wondering something.
The Riemann hypothesis essentially states that provided there are no non-trivial zeroes of the Riemann Zeta function that fall off the critical line, the error between the estimate of the number of primes up to x, x/ln(x), will stay within a given bound.
What I'm curious about is that it's been shown that the first however many zeroes are all on the trivial line, and the first however many primes are within this bound.
Does that imply that if there were to be an off-line zero, it would affect a specific prime, like the Xth prime where X is some huge number will be out of error bounds because the Yth zero where Y is another huge number is off the line? Or does it affect the structure of primes as a whole in some way that simply wouldn't be clear by looking at the first few trillion or whatever?
Basically, I'm asking if a non critical line Riemann zero exists, does the position of that zero have anything to do with the position of where the primes drop out of the error bound? If not, how exactly would it affect the primes?