Changes

: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)

::The talk-to-substance ratio is very high in the above replies, and that's not a good sign.  Complex analysis does rely on an assumption that the there is a unique, algebraically manipulable square root of negative one.  It's a unique pair, obviously, and it's a sign of desperation that some above looked for a way to deny this.

::The talk-to-substance ratio is very high in the above replies, and that's not a good sign.  Complex analysis does rely on an assumption that there is a unique, algebraically manipulable square root of negative one.  It's a unique pair, obviously, and it's a sign of desperation that some above looked for a way to deny this.

::Proofs that rely on this assumption are obviously less favored than proofs that do not, and hence the important concept of an [[elementary proof]].  Deny its importance all you want, but the beauty of logic is that it doesn't matter who accepts or denies it.  The logic remains as truthful regardless.--[[User:Aschlafly|Andy Schlafly]] 11:26, 27 August 2009 (EDT)

::Proofs that rely on this assumption are obviously less favored than proofs that do not, and hence the important concept of an [[elementary proof]].  Deny its importance all you want, but the beauty of logic is that it doesn't matter who accepts or denies it.  The logic remains as truthful regardless.--[[User:Aschlafly|Andy Schlafly]] 11:26, 27 August 2009 (EDT)