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Apr 5, 2011 at 9:30 comment added user9072 @Seva: From the MR review of Daboussi's paper (written 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." Here is a link to the original paper of Daboussi math.jussieu.fr/~allouche/20060929144515189.pdf (And then there are the other devlopments improving on the error term, Diamon-Steinig and so on.)
Apr 5, 2011 at 7:44 comment added Seva Somehow, I thought there is essentially just one "elementary" proof known, that due to Erdős and Selberg; do you mean, the proof you mention is substantially different?
Apr 5, 2011 at 0:06 history answered Gerry Myerson CC BY-SA 2.5