Why? Well, this is because a formula or expression that receives an integer at its input and always returns a prime number at its output, regardless of the numerical size, is humanly and technologically impossible. Prime numbers are infinite, and the larger they are, the more difficult it is to predict them without making mistakes, therefore the formula is not linear and its slope changes as the numbers grow. Due to this, since the entire number space generated by the primes cannot be known, Riemann's postulate cannot be verified either.
Not to mention the fact that the Riemann hypothesis isn’t about producing prime numbers, but a statement about their distribution. Its validity doesn't rely on actually writing down all the roots or all the primes.
However if one has never learned about proof by induction, it's easy to get confused.
Dear Peter,
What you expose could be considered as a contradiction. If the Riemann hypothesis for non-trivial zeros has not been proved yet, then how could we know the distribution of all primes, or why?.
It's not about confusion, it's about facts. Riemann's postulate still lacks a formal proof, with or without the method of Induction.
Yes, everything is very interesting analytically speaking, but its distribution does not contemplate a variable slope that precisely changes when we work on the furthest primes towards infinity. And this is the reason that we have to verify each of the predictions, there is no such thing as complete certainty at the far end of the distribution. Riemann's zeta function can be written according to Euler as a function of the series of prime numbers, that is why I suggest that empirically Riemann is not totally certain that his hypothesis is fulfilled in the primes existing in it infinite. That is, the distribution of primes is not unique, it constantly changes as we move towards larger numbers, so the predictions are neither safe nor exact. I don't know if I'm expressing myself in the most appropriate way.