Humans have discovered independence proofs, e.g. Paul Cohen’s 1963 proof that the continuum hypothesis is independent of ZFC. I can’t see any reason in principle why a computer couldn’t do the same.

If the Riemann hypothesis is independent of ZFC, and there exists a proof of that independence which is of tractable length, then in principle if a human could discover it, why couldn’t a sufficiently advanced computer system?

Of course, it may turn out either that (a) Riemann hypothesis isn’t independent of ZFC (what most mathematicians think), or (b) it is independent but no proof exists, or (c) the shortest proof is so astronomically long nobody will ever be able to know it

> The incompleteness theorem for example is a meta-mathematical statement about the limits of axiomatic systems that can not be discovered with axiomatic systems alone.

We have proofs of Gödel‘s theorems. I see no reason in principle why a (sufficiently powerful) automated theorem prover couldn’t discover those proofs for itself. And maybe even one day discover proofs of novel theorems in the same vein

Riemann is so widely believed to be true that there are entire branches of mathematics dedicated to seeing what cool things you can learn about primes/combinatorics etc by taking Riemann to be true as an assumption.