TITLE: Complex and Elementary Proofs in Number Theory
QUESTION [19 upvotes]: 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.

REPLY [3 votes]: Here is a recent paper by Dimitris Koukoulopoulos, which obtains the strongest known form of the Prime Number Theorem without heavy use of complex analysis. One can find the symbol $i$ in the paper, but it does not rely on the analytic continuation of the zeta function.
This is a direct extension of the "pretentious" methods of Granville and Soundararajan, mentioned in an answer by Thomas Riepe.