ANALYTIC NUMBER THEORY
Arithmetical functions, Primes in Arithmetic Progressions, Prime number theorem for arithmetic progressions and zeros of Dirichlet L-functions, Bombieri-Vinogradov theorem, Equidistribution, circle method and applications (ternary goldbach in mind), the Large Sieve and applications, Brun's theorem on twin primes.
(Further topics if time permits: more on sieves, automorpic forms and L-functions, Hecke's L-functions for number fields, bounds on exponential sum etc.)
Basic of number theory, compleX analysis, preferably some familiarity with MA 398 (Introduction to Anaytic number theroy).
Multiplicative Number Theory by H. Davenport. Springer Graduate Texts in Mathematics 74.
Problems in Analytic Number Theory by M. Ram Murty. Springer Graduate Texts in Mathematics 206.
Analytic Number Theory by H. Iwaniec and E. Kowalski. AMS Colloquium Publ. 53.