Context: I am a junior math major and am hoping to go to grad school after next year for a PhD. I have completed most of the standard undergraduate courses and have been consistently most interested by number theory. I've taken three courses in number theory; the first using Niven's text, the second using Ireland and Rosen, and the third using Montgomery and vaughan's Multiplicative Number Theory.
Next year, I have room for four independent study courses. One of them will likely be on (introductory) algebraic geometry but with the other three, I wish to gain some breadth within number theory to find out what I am most passionate about.
Questions: What are the popular branches of number theory being actively researched today and what are some good introductory texts in each of these?
Best Answer
Gaps between primes is certainly active at the moment so I'd recommend Harman's book Prime-Detecting Sieves. Generally Iwaniec and Kowalski's book Analytic Number Theory and Hardy & Wright's book Introduction to the Theory of Numbers. Also I'd say that the circle method is fairly commonly used at the moment and Bob Vaughan has a great book on it entitled the Hardy-Littlewood Method.
Also sumset problems seem to be big at the moment and Tao & Vu's book Additive Combinatorics is a great read! Ben Green has some good expository papers online on the topic as well.