Bateman and Diamond's Analytic Number Theory is a graduate level textbook in the subject. The primary prerequisites are beginning graduate level courses in complex analysis and real analysis. A prior course in elementary number theory would be useful but not necessary.
The text covers standard topics such as the distribution of primes numbers, including Dirichlet's theorem on primes in arithmetic progressions, and basic properties of the Riemann zeta-function and Dirichlet L-functions. The authors give both elementary and analytic proofs of the Prime Number Theorem. A less standard, but very interesting topic, is a thorough treatment of Dirichlet convolution, including a discussion of convergence of infinite convolution products. The authors present the Weiner-Ikehara approach to the Prime Number Theorem, along with generalizations that lead to the asymptotic formula for number of sums of two squares.
The book also includes a nice introduction to sieve methods. The Brun-Hooley sieve is presented in the general formulation developed by the author's colleagues K. Ford and H. Halberstam. This sieve is "revisionist history"; this is not a sieve that Brun developed, but it is a nice variation that Hooley found that is very much in the spirit of Brun's approach. The authors also present the large sieve using the Beurling-Selberg extremal function approach. Finally, the authors present some applications of sieves, including the Brun-Titchmarsh upper bound for primes in intervals and upper bounds for the number of twin primes.
Overall, this is a nice well-written book with plenty of material for a one-year graduate course. It would also make nice supplementary reading for a student or researcher learning the subject.
S.W. Graham is currently a Professor of Mathematics at Central Michigan University. From 1995 to 1998, he was a program director in the Algebra and Number Theory Program at the National Science Foundation. Before that, he held faculty positions at Michigan Tech, the University of Texas, and California Institute of Technology. He received his Ph. D. at the University of Michigan under the direction of Hugh Montgomery in 1977. He can be reached at email@example.com.