Regarding the summary of proofs of Prime Number Theorem I have read the proof of Prime Number Theorem given in Introduction to Analytic Number Theory, Apostol.
Now I want to know the following, I would be glad if someone can help
1-Whose proof did Tom Apostol follow in this book on Analytic Number Theory which can be found here?
Was this proof given by Riemann? 
2-What are all the other proofs that have been given for PNT?
3-Which proof do you think is worth studying?
 A: First of all, Riemann did not give a proof of the Prime Number Theorem.  He gave a sketch of an approach to proving it, and eventually his ideas did work out, but the necessary ingredients from the zeta-function turned out to be much less than was at first believed.  Ultimately all you need to know about the zeta-function outside its initial half-plane of convergence ${\rm Re}(s) > 1$ is that it extends analytically to the line ${\rm Re}(s) = 1$ except for a simple pole at $s = 1$.  No growth conditions for the zeta-function that line are needed; earlier proofs required such information (and a whole lot more, but as I already wrote, over time the proof was greatly simplified).  
To address your questions:


*

*Apostol's proof is similar to the proof in Ingham's book on the distribution of prime numbers (from the 1930s).  Both proofs involve working with $\psi_1(x) = \int_0^x \psi(y)\,dy$ instead of $\psi(x)$ directly. (Davenport writes in his book "Multiplicative Number Theory" that the idea of working with $\psi_1(x)$ instead of $\psi(x)$ goes back to de la Vallée Poussin's proof of PNT in 1896.) 

*All other proofs?!? It's more realistic to ask for some other proofs. Please clarify what it is you really want to know. Maybe you might like to read Narkiewicz's book "The Development of Prime Number Theory from Euclid to Hardy and Littlewood". He discusses what went into the first proofs of PNT by Hadamard and de la Vallée Poussin as well as Landau and the Tauberian approach, which is how the asymptotic relation in PNT is often proved today, especially if you are not interested in error terms but just a plain asymptotic relation. 

*If you just care about PNT as a pure asymptotic statement, then read D. J. Newman's proof.  It's from 1980 and is probably the simplest proof currently available.  You can find it in Jameson's book "The Prime Number Theorem" and many recently published analytic number theory books. 
A: Just as a supplement to 3. There is a famous article by Don Zagier about Newman's Short Proof of the Prime Number Theorem, which certainly is worth studying. It has $4$ pages.
