A good reference to begin analytic number theory

I know a little bit about basic number theory, much about algebra/analysis, I've read most of Niven & Zuckerman's "Introduction to the theory of numbers" (first 5 chapters), but nothing about analytic number theory. I'd like to know if there would be a book that I could find (or notes from a teacher online) that would introduce me to analytic number theory's classical results. Any suggestions?

Thanks for the tips,

Solution 1:

I'm quite partial to Apostol's books, and although I haven't read them (yet) his analytic number theory books have an excellent reputation.

Introduction to Analytic Number Theory (Difficult undergraduate level)

Modular Functions and Dirichlet Series in Number Theory (can be considered a continuation of the book above)

I absolutely plan to read them in the future, but I'm going through some of his other books right now.

Ram Murty's Problems in Analytic Number Theory is stellar as it has a ton of problems to work out!

Solution 2:

• If you haven't read the chapter on Dirichlet's theorem on primes in arithmetic proression in Serre's Course in arithmetic, I highly recommend that you do. You can read it independently of what came before.

• I liked the book of Ayoub when I was a student. My memory is that it is somewhere between a textbook and a monograph, and that it covers lots of fundamental topics, such as partitions, Dirichelt's theorem, the circle method, and so on. I found it compelling enough that I failed an English course because I spent all my time reading the book instead of writing the required essay.