TITLE: Quantitative and elementary proofs of the Prime Number Theorem
QUESTION [10 upvotes]: I would like to know two things: one, whether the best quantative bounds in the Prime Number Theorem are still basically those given by the Vinogradov-Korobov zero-free region? and two, whether there are any elementary proofs substantially different from the Erdős/Selberg proofs?
I realise this is probably trivial to answer for experts, but it seems to be hard to find clear statements in the literature for non-experts.

REPLY [4 votes]: The proof that is broadly speaking elementary, in the sense that it does use the analytic continuation of Dirichlet L-fucntions, which gives the same error term as Korobov-Vinogradov zero free region was given a few years ago by Koukouloupolos, see http://www.dms.umontreal.ca/~koukoulo/documents/publications/pnt.pdf