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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.

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    $\begingroup$ The following MO question discusses an elementary proof by Daboussi which is different from Erdos/Selberg: mathoverflow.net/questions/219705/… $\endgroup$
    – Felipe Voloch
    Commented Oct 1, 2015 at 13:36
  • $\begingroup$ @FelipeVoloch This seems to be completely hidden (the many exposes on the "elementary proof of the PNT" published in English seem to ignore this completely. $\endgroup$
    – Igor Rivin
    Commented Oct 1, 2015 at 15:28
  • $\begingroup$ @IgorRivin this strikes me as an exaggeration. (check the citations on Google Scholar for instance.) Of course it was only found in the first half of the 80s so prior expositions can hardly account for it. $\endgroup$
    – user9072
    Commented Oct 1, 2015 at 16:24

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The best known error term in PNT is to my knowledge Big-Oh of $$x\exp \left(-C \frac{ (\log x)^{\frac35}}{(\log \log x)^{\frac15}} \right),$$ and for $C$ one can take $-0.2098$, given in:

K. Ford. Vinogradov’s integral and bounds for the Riemann zeta function. Proc. London Math. Soc., 85(3):565–633, 2002.

Also see "Updating the error term in the prime number theorem" by Tim Trudgian http://arxiv.org/abs/1401.2689, especially note the footnote on page 4.

Thus there was some progress yet the factor $\frac35$, already known from Vinogradov-Korobov, was not improved up to now.

On the elementary proof: "Substantially different" is always hard to answer, but meanwhile there are elementary proofs that give an error term of basically the same quality as Vinogradov-Korobov, which was not at all the case for the original elementary proof. Thus, there was quite some progress there.

For an overview of elementary proofs you could consult this earlier MO question: Prime Number Theorem w/o Complex Analysis

In particular note that not all proofs are based on Selberg's formula. For example as mentioned in a comment by Voloch the one by Daboussi is not, as for example explained in its MR review by Diamond: "This paper gives an elementary proof of the PNT that is remarkable in that it makes no use of Selberg's now famous formula."

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A nice survey is given by Hildebrandt, in his 2013 UIUC lecture notes. The answer to your first question is "yes", and to the second question, depends what you mean by "substantially different" - all proofs seem to be based on Selberg's inequality, but as you will see from Hildebrandt's notes, some get better error terms than others. A nice account of the elementary proof is Levinson's 1969 Monthly article.

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  • $\begingroup$ I am pretty sure to have seen elementary proof with better then what is said in those notes. Also the notes you link to do not account for improvements on the constant for the general question. At least not in the tables at the start. $\endgroup$
    – user9072
    Commented Oct 1, 2015 at 11:42
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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

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