I know about
"Simple analytic proof of the prime number theorem" Newman, 1980
However, is there a proof of the Prime Number Theorem without the use of complex analysis? (Real analysis is fine).
Thanks!
I know about "Simple analytic proof of the prime number theorem" Newman, 1980 However, is there a proof of the Prime Number Theorem without the use of complex analysis? (Real analysis is fine). Thanks! 


http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf explains the classic proof in context (there is what amounts to a priority dispute). 


Another exposition of an elementary proof (that is, a proof not using complex analysis) is in Gerald Tenenbaum and Michel Mendes France, The Prime Numbers and Their Distribution, which is Volume 6 of the Student Mathematical Library, published by the American Mathematical Society. The proof they give is due to Daboussi, from 1984. 


If you just want $\pi(n) = \Omega \left( \frac{n}{\log n} \right)$, good enough for many applications, here is a quick proof: The highest power of a prime $p$ dividing $2n \choose n$ is at most $2n$  you get at most one more factor of $p$ in the numerator than denominator for each power $p^i \leq 2n$. This tells you that ${2n \choose n} \leq (2n)^{\pi(2n)}$. So $\pi(2n) \geq \frac{\log_2 {2n \choose n}}{\log_2 (2n)} \geq \frac{n}{\log_2 (2n)}$. 


There is a terrific exposition of the elementary proof by Terry Tao, available as the file prime.dvi here. A more traditional exposition is available in Edwards's book Riemann's zeta function. 


A nice exposition of an Erdos/Selbergtype elementary proof is given by Levinson in Amer. Math. Monthly 76 (1969) 225–245. The proof by Daboussi as written up by Tenenbaum and MendesFrance was already mentioned. Yet another one is due to Hildebrand in Mathematika 33 (1986) 23–30. 

