This book provides an introduction to four central problems in analytic number theory. These are (1) the problem of estimating the number of integerpoints in planar domains (2) the problem of the distribution of prime numbers in the sequence of all natural numbers and in arithmetic progressions (3) Goldbach's problem on sums of primes, and (4) Waring's problem on sums of k-th powers. To solve these problems, one uses the fundamental methods of analytic number theory: complex integration, I.M.Vinogradov's method of trigonometric sums, and the circle method of G.H.Hardy, J.E.Littlewood, and S.Ramanujan. There are numerous exercises at the end of each chapter. These exercises either refine the theorems proved in the text, or lead to new ideas in number theory. The author also includes a section of hints for the solution of the exercises. The mathematical prerequisites for this volume are undergraduate courses in number theroy, mathematical analysis, and complex variables. The book would be an excellent text for a one or two semester course in analytic number theory for advanced undergraduates or graduate students.

Springer-Verlag has invited me to bring out my Selected Works. Being aware that Springer-Verlag enjoys high esteem in the scientific world as a reputed publisher, I have willingly accepted the offer. Immediately, I was faced with two problems. The first was that of acquaint ing the reader with the important stages in my scientific aetivities. For this purpose, I have included in the Selected Works eertain of my early works that have greatly influeneed my later studies. For the same reason, I have also in cluded in the book those works that contain the first, erude versions of the proofs for many of my basic theorems. The second problem was that of giving the reader the best possible opportunity to familiarize himself with the most important results and to learn to use my method. For this reason I have included the later improved versions of the proofs for my basic results, as weil as the monographs The Method of Trigo nometric Sums in Number Theory (Seeond Edition) and Special Variants of the Method of Trigonometric Sums."