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To elaborate on my comment, I really do think this won't be resolved until RH itself is resolved, or at least until new tools are brought onto the scene, because it's so far from what we know about the zeros by current technology. All the theorems we have about zeros of the zeta function - e.g., the zero-free region, the zero-density theorems, Montgomery's conditional results on pair correlation, Selberg's theorems on $S(t)$ - are either incredibly weak compared to waht what we expect to be true (Iwaniec often refers to zero-density theorems as "estimates for the cardinality of the empty set"), or are theorems "in the large", that is to say they deal in whole masses of zeros as opposed to individual zeros. Conventional harmonic analysis is simply unable to grasp individual zeros, for reasons of the uncertainty principle.

The closest reference I know to your question is a paper of Bombieri, "Remarks on Weil's quadratic functional in the theory of prime numbers", where he shows that if the RH is false for only a finite number of zeros, then very odd things follow...

Edit: Thinking about this question a bit more has led me to a conjecture seems quite a bit weaker than anything approaching RH and which essentially implies the impossibility of finitely many failures.

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To elaborate on my comment, I really do think this won't be resolved until RH itself is resolved, or at least until new tools are brought onto the scene, because it's so far from what we know about the zeros by current technology. All the theorems we have about zeros of the zeta function - e.g., the zero-free region, the zero-density theorems, Montgomery's conditional results on pair correlation, Selberg's theorems on $S(t)$ - are either incredibly weak compared to waht we expect to be true (Iwaniec often refers to zero-density theorems as "estimates for the cardinality of the empty set"), or are theorems "in the large", that is to say they deal in whole masses of zeros as opposed to individual zeros. Conventional harmonic analysis is simply unable to grasp individual zeros, for reasons of the uncertainty principle.

The closest reference I know to your question is a paper of Bombieri, "Remarks on Weil's quadratic functional in the theory of prime numbers", where he shows that if the RH is false for only a finite number of zeros, then very odd things follow...