I think basic is on the right track. The two big classical theorems in analytic number theory whose classical proofs use some complex analysis are Dirichlet's Theorem on primes in arithmetic progressions and the Prime Number Theorem. (It is also useful to learn about the combination of the two: the Prime Number Theorem for Arithmetic Progressions.)
For the former, I can recommend my own lecture notes:
The second part is explicitly a digested version of the proof Serre presents in his Course in Arithmetic. I don't have a similarly canonical reference to give you for the proof of the Prime Number Theorem (i.e., I don't have any notes on it!), but it can be found in many analytic number theory books, for instance in Apostol's Introduction to Analytic Number Theory or, as has been suggested in the comments, Davenport's Multiplicative Number Theory or G.J.O. Jameson's The Prime Number Theorem.