The Prime Number Theorem was originally proved using methods in complex analysis. Erdos and Selberg gave an elementary proof of the Prime Number Theorem. Here, "elementary" means no use of complex function theory.
Is it possible that any theorem in number theory can be proved without use of the complex numbers?
On the one hand, it seems a lot of the theorems using in analytic number theory are about the distributions of primes. Since the Prime Number Theorem has an elementary proof, this might suggest that elementary proofs exist in other cases.
On the other hand, the distribution of primes is intimately related to the zeros of the Riemann Zeta function. Perhaps the proofs of other statements in analytic number theory require more direct references to the Riemann Zeta function.
This topic is more of a fascination for me, as I am not a number theorist. I would be interested if there are other examples of elementary proofs of theorems originally proved with complex analytic methods.
Best Answer
Yes, there is a theorem to this effect by Takeuti given in his book "Two applications of logic to mathematics". He shows roughly that complex analysis can be developed in a conservative extension of Peano arithmetic.