Additive Number Theory

The purpose of this book is to describe the classical problems in additive number theory, and to introduce the circle method and the sieve method, which are the basic analytical and combinatorial tools to attack these problems. This book is intended for students who want to learn additive number theory, not for experts who already know it. The prerequisites for this book are undergraduate courses in number theory and real analysis.

Customer Reviews:

Clear coverage of Waring's Problem and Goldbach Conjecture

By A Customer - April 21, 2002

This book gives clear and complete proofs of Waring's Problem (that every positive integer is the sum of a bounded number of nth powers) and of all current results in the Goldbach Conjecture (Brun's theorem that the sum of the reciprocals of the twin primes converges, Vinogradov's theorem that every large odd integer is the sum of three primes, and Chen's theorem that every large even integer is the sum of a prime and and number that is either prime or the product of two primes). The focus of the book is these specific problems; it develops many general methods while attacking these problems, but does not develop the general methods for their own sake. The book assumes a little prior knowledge of analysis and number theory, and it quotes a few advanced results (for example, the Bombieri-Vinogradov theorem on primes in arithmetic progressions) that are proved in Davenport's book "Multiplicative Number Theory", but otherwise it is a complete exposition of these two problems.The best... read more

Advanced graduate level text in additive number theory

By Stuart-Little - October 21, 2007

Advanced graduate level text in additive number theory, covers the classical bases. This book is the first comprehensive treatment of the subject in 40 years. If the topic of additive number theory interests you, then this is the book to get as there is no comparable (single) book available. Requires a solid understanding of complex analysis. Note, a nice introduction to additive number theory can be found in Hardy and Wright's Introduction to Number Theory.

Some highlights: 1) Chen's theorem that every sufficiently large even integer is the sum of a prime and a number that is either prime or the product of two primes. 2) Brun's sieve for upper bound on the number of twin primes. 3) Vinogradov's simplification of the Hardy, Littlewood, and Ramanujan's circle method.