Reddit reviews Introduction to Analytic Number Theory

His number theory books, Introduction to Analytic Number Theory and Modular Forms and Dirichlet Series in Number Theory, are some of the best in their respective subjects. Good to learn from and good for references.

Set Theory:

Naive Set Theory

Number Theory:

Elementary Number Theory

Introduction to Analytic Number Theory

A Classical Introduction to Modern Number Theory

Topology:

Topology

Introduction to Topological Manifolds

I would strongly recommend Apostle's intro to analytic number theory. It doesn't assume any prior knowledge of number theory, but does move a bit fast.

Some folks would probably suggest you get a book solely dedicated to elementary number theory, but the danger is that you can get bogged down in in the details and never get to the good stuff.

There's a couple options. You could pick up a basic elementary number theory book, which will have basically no prerequisites, so you'll be totally fine going into it. For instance Silverman has an elementary number theory book that I've heard great things about. I haven't read most of it myself, but I've read other things Silverman has written and they were really good.

There's a couple other books you might consider. Hardy and Wright wrote the classic text on it, which I've heard still holds up. I learned my first number theory from a book by Underwood Dudley which is by far the easiest introduction to number theory I've seen.

Another route you might take is that, since you have some background in calculus, you could learn a little basic analytic number theory. Much of this will still be out of your reach because you haven't taken a formal analysis class yet, but there's a book by Apostol whose first few chapters really only require knowledge of calculus.

If you decide you want to learn more number theory at that point, you're going to want to make sure you learn some basic algebra and analysis, but these are good places to start.

Introduction to Analytic Number Theory by Apostol is a great introduction to analytic number theory. This would be a great way to tie together the number theory, combinatorics, and calculus that you've seen so far.