I had decided to study and understand the proof of prime number theorem. I had tried to approximate the value of $\frac{\pi(x)}{x}$ and have seen some videos which gives the intuition about the approximation.
So I want to ask for references for a beginner. I mean it should be either understandable easily or with a bit of hardwork. It shouldn't be very difficult. The thing is to get comfortable with this new topic analytic number theory, I first need to go through an easy proof.
$1 $ More question I would like to ask is that-Isn't Tom Apostol's Book on analytic number theory has a nice and systematic analytic proof of that theorem. What are the topics I need to study to understand the proof of prime number theorem by Tom Apostol?
P.S- I had already done courses in Number Theory, Real Analysis, Complex Analysis
At the 100th birthday of the prime number theorem Don Zagier published Newman's Short Proof of the Prime Number Theorem in the American Mathematical Monthly. It's only three pages, without the Historical Remarks. You can embed it in a first course of complex analysis.
It's in the October issue of American Mathematical Monthly 104 (1997), 705–708.