# Talk:Prime Number Theorem

## Rewrite

Okay, the old proof wasn't finished, so I've rewritten the article entirely with the proof of D.J. Newman. I think it would be good to have several different proofs, though, so if somebody else knows how the other proof goes, I think it would be good to add that, as well. At some later point, I might read the Selberg-Erdős proof, and write that up as well, or maybe somebody else will. The old version of the article is <url>Wiki/index.php?title=Prime_Number_Theorem&oldid=10287 here</url>.

I've also added a bunch of links to papers online. Fortunately, I could find copies of all the papers outside of JSTOR, so that everybody can see them.

I think there should maybe be more discussion of the logarithmic integral, as this seems to be in fact a better approximation of $\pi(x)$. I don't know much about it, though, so perhaps somebody else should do this. Perhaps it should go into a separate article, since it's not really part of the prime number theorem. —Boy Soprano II 06:10, 10 April 2009 (UTC)

Oh, also, I couldn't find an online copy of de la Vallée Poussin's orginal manuscript, so if anybody knows of a copy of that, it would be nice to have for completeness and historical interest. It might also be interesting to have some of the successive improvements on the proof. —Boy Soprano II 06:16, 10 April 2009 (UTC)

This proof is much harder than the proof using the zeta function. To do that, you just have to show that there are no zeros on the one line. --ComplexZeta 12:48, 29 June 2006 (EDT)

Hmmm... The fact that zeta-function has no zeroes with $\Re z=1$ will be used here for sure but it is not the end of the story: to the best of my knowledge, to finish, one has to use either Riemann's explicit formula for $\pi(x)$, or some complex analysis trick, or some Tauberian theorem. I was inclined to use the last approach because I find it a bit more natural than the other two. But, perhaps, you know some clever shortcut I am unaware of. If so, I'll be most grateful if you write the proof you know. --Fedja 13:15, 29 June 2006 (EDT)