I learned analytic number theory from his text book. Probably unlike most other HN participants, that is where I learned big-oh notation, in its original context: measuring the growth of arithmetic functions. I distilled what I learned into a short little undergrad paper:

http://jordi.platinum.linux.pl/dirichlet-primes.pdf

Thank you, Tom Apostol, for writing a book on analytic number theory for me to learn from.