I tried to understand Terence Tao's Analytic Number Theory Papers. For example, this paper, Every Odd Number Greater Than 1 is The Sum of at Most Five Primes.
Which books shall I read to prepare myself to understand those papers ? Maybe there will be a sequence of books ?
I do not have analytic number theory background. The only number theory book I read is Hardy and Wright, Introduction to the Theory of Numbers. My background is in Algebra and Complex Analysis. (I also read a few chapters of Apostol, but I felt it is not enough to understand Terence Tao's papers).
Best Answer
Dr. Tao's papers often evidence a good understanding and use of analytic number theory, additive number theory, and the circle method. So if you want to read your way towards understanding the papers, this would be a good trio to go towards.
For analytic number theory, you want to understand some algebra and some complex analysis. With these, you could understand Apostol's Introduction to Analytic Number Theory and then perhaps Montgomery and Vaughan's Multiplicative Number Theory.
For additive number theory, you want to understand algebra, real analysis, and probability theory. The best book to understand Tao's papers is Tao and Vu's Additive Combinatorics. This is pitched really high, and it's extremely likely that one would need to backtrack and fill in gaps while attempting to go through it.
For the circle method itself, I would recommend Vaughan's The Hardy-Littlewood Method. (This is another name for the circle method). Alternately, since every application of the method is so different, you might just try to understand it directly from the papers themselves.
A good supplement to the last two books, and a good book in general, is Nathanson's Additive Number Theory, the Classical Bases which approaches some of the classical results of additive number theory, sometimes using the circle method.