The Hardy-Littlewood method / R. C. VaughanType de document : MonographieCollection : Cambridge tracts in mathematics, 80Langue : anglais.Pays : Grande Bretagne.Éditeur : Cambridge : Cambridge University Press, 1981Description : 1 vol. (172 p.) ; 23 cmISBN : 9780521234399.ISSN : 0950-6284.Bibliographie : Bibliography: p. -168. Index.Sujet MSC : 11P55, Additive number theory; partitions, Applications of the Hardy-Littlewood method
11P05, Additive number theory; partitions, Waring's problem and variants
11P32, Additive number theory; partitions, Goldbach-type theorems; other additive questions involving primes
11D75, Number theory - Diophantine equations, Diophantine inequalities
11Pxx, Number theory - Additive number theory; partitionsEn-ligne : Zentralblatt | MathSciNet | Edition 1997
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The admirable book under review makes a notable introduction to the Hardy-Littlewood method, which sometimes is called the "circle method'', and its more recent developments in analytic number theory.
The genesis of the method is to be found in a paper of Hardy and Ramanujan in 1918 and the early essential work on the method appeared in a series of papers entitled "Some problems of Partitio Numerorum'' by Hardy and Littlewood published during the period 1920–1928. Historical progress shows that during the past almost sixty years the method has had very striking and successful applications to Waring's problem, Goldbach's problem and many other additive problems in number theory. One of the remarkable examples is that Vinogradov refined the method to show that every large odd number is the sum of three odd primes.
The book consists of eleven chapters. The historical background of the method is given in Chapter 1. Chapters 2,4,5,6 and 7 are devoted to the study of the G(k) in Waring's problem. Contributions to the method due to Davenport, Hua and Vinogradov are very well presented in these chapters, to obtain first the simplest bound and then Vinogradov's bound for G(k). Chapter 3 deals with Goldbach's problem. This chapter contains the author's elegant and short proof of the famous Vinogradov lemma for the trigonometric sum. Chapters 8,9,10 and 11 are concerned with recent developments of the method in some important additive problems, such as the author's results on a ternary additive problem, Birch's theorem on homogeneous equations, Roth's theorem on arithmetic progression, Davenport and Heilbronn's theorem on Diophantine inequalities. At the end of each chapter some interesting and usually difficult exercises are given. Following the last chapter, there are an index combining subjects and authors and a comprehensive bibliography consisting of about 400 references.
The author has himself made many significant contributions to the method; his book is therefore particularly welcome. Although the reader is expected to be familiar with elementary number theory the book is essentially self-contained and very readable. The contents of most chapters of the book can be used in advanced courses in analytic number theory. The author's style is elegant and the exposition is lucid throughout because of his deep insight into this field. The book will certainly interest both specialists and postgraduate students and become an excellent standard reference on the Hardy-Littlewood method. (MathSciNet)
Bibliography: p. -168. Index