There is a famous circular argument for the Prime Number Theorem (PNT). It turns out that there exists an infinite sequence of elementary-to-prove Chebyshev-type estimates that taken together imply PNT. Unfortunately, the collective existence of all these proofs seems to require the PNT, so one must work hard a la Selberg and Erdos for an elementary proof. See Harold Diamond, Elementary methods in the study of the distribution of prime numbers, Bull. Amer. Math. Soc. N. S. 7 (1982), 553-589.
Now on the traditional view, circular proofs simply have no value at all. Yet one feels perhaps that the example in the previous paragraph has something to say. Just an illusion? Or does there exist a foundational framework where circular proofs of this special sort enjoy bona fide status?
Just to riff a little more, imagine that the Goldbach conjecture turns out independent of PA (or some other, perhaps weaker, axiom system for arithmetic). The truth of the Goldbach conjecture (required for independence!) would then imply the existence of a proof (trivial!) for any given even number that that one even number equals the sum of two primes. Now that obviously isn't very interesting compared to the example in the first paragraph, and perhaps the difference has something to do with the greater quantifier complexity of PNT?
As a side question, are there any similar stories in the lore of the Riemann Hypothesis? For example, does RH imply the existence of an infinite sequence of zero-free region proofs of a known form that collectively amount to RH?
EDIT: Lest this come up repeatedly, let me expand upon a remark I made in a comment below:
I would be interested to know if RH implies the existence of proofs of a known form in a known system within which one does not assume RH, such that the conclusions of all these proofs conjunctively yield RH.
Actually the answer to my question is "yes" though I find my own example unsatisfying (rather like the Goldbach example):
We can check zeros up to a given magnitude rigorously by known (non-trivial) numerical techniques. RH predicts these tests will come out positive, but of course the tests don't rely on RH. If we know they all come out positive, that's RH. I find this unsatisfying because the little proofs approximate the whole of RH so badly (compared to how the Chebyshev estimates really do make one feel one has PNT for all practical purposes). Now a family of zero-free regions that union up to $\sigma >1/2$ where each new zero-free region had strong arithmetic consequences, that would seem interesting.