What complex analysis must I know to understand D.J. Newman's proof of the Prime Number Theorem?

Question

The title sums it up, but let me flesh out my question. First off, I am referring to a simplification of Newman's proof given by Zagier (link), and expanded for clarity here. Now, I am fairly comfortable with calculus, real analysis, and some number theory. However, I have never studied complex analysis (though I understand some basics of complex variables/functions, as I have read the relevant chapters in Spivak's calculus book). What I would like, is to know precisely which topics of complex analysis I need to cover to be able to understand the aforementioned proof; even better, a recommendation of a book (or books) and which chapters are essential to understand the proof argument would be excellent.

Answer 1

You need to know about residues and analytic continuation applied to the zeta function.

The book Complex Variables by Ash and Novinger ends with an exposition of a proof of the Prime Number Theorem along the lines you want:

Our approach to the prime number theorem has been along traditional lines, but at this stage we will apply D.J. Newman’s method [...] as modified by J. Korevaar. (section 7.3)

There is also the exposition at end of Complex Analysis by Bak and Newman. Yes, that Newman.