:An unrelated issue is that there seem to be two notions of elementary proof used in this article, and I'm not sure how to reconcile them. Selberg's proof of PNT is magnificently complicated, though it avoids the methods of analytic number theory. It's not a proof that I would say "cannot be improved by expressing it in simpler form" -- indeed the analytic proofs are far simpler. It's true that at heart these have a different method, but it's my understanding that a touch of complex analysis can simplify the Selberg argument while leaving its fundamental insights intact. --[[User:MarkGall|MarkGall]] 00:24, 27 August 2009 (EDT) | :An unrelated issue is that there seem to be two notions of elementary proof used in this article, and I'm not sure how to reconcile them. Selberg's proof of PNT is magnificently complicated, though it avoids the methods of analytic number theory. It's not a proof that I would say "cannot be improved by expressing it in simpler form" -- indeed the analytic proofs are far simpler. It's true that at heart these have a different method, but it's my understanding that a touch of complex analysis can simplify the Selberg argument while leaving its fundamental insights intact. --[[User:MarkGall|MarkGall]] 00:24, 27 August 2009 (EDT) |