There have been two earlier Cambridge Tracts that have touched upon the Hardy-Littlewood method， namely those of Landau， 1937， and Estermann， 1952. However there has been no general account of the method published in the United Kingdom despite the not inconsiderable contribution of English scholars in inventing and developing the method and the numerous monographs that have appeared abroad. The purpose of this tract is to give an account of the classical forms of the method together with an outline of some of the more recent developments. It has been deemed more desirable to have this particular emphasis as many of the later applications make important use of the classical material.本書為英文版。
作者簡介
暫缺《Hardy-Littlewood 方法（第2版）》作者簡介
圖書目錄
Contents
Preface
Preface to second edition
Notation
Introduction and historical background
1.1 Waring's problem
1.2 The Hardy-Littlewood method
1.3 Goldbach's problem
1.4 Other problems
1.5 Exercises
The simplest upper bound for G(k)
2.1 The definition ofmajor and minor arcs
2.2 Auxiliary lemmas
2.3 The treatment of the minor arcs
2.4 The major arcs
2.5 The singular integral
2.6 The singular series
2.7 Summary
2.8 Exercises
Goldbach's problems
3.1 The ternary Goldbach problem
3.2 The binary Goldbach problem
3.3 Exercises
The major arcs in Waring's problem
4.1 The generating function
4.2 The exponential sum S(q, a)
4.3 The singular series
4.4 The contribution from the major arcs
4.5 The congruence condition
4.6 Exercises
Vinogradov's methods
5.1 Vinogradov's mean value theorem
5.2 The transition from the mean
5.3 The minor arcs in Waring's problem
5.4 An upper bound for G(k)
5.5 Wooley's refinement of Vinogradov's mean
value theorem
5.6 Exercises
Davenport's methods
6.1 Sets ofsums of kth powers
6.2 G(4) = 16
6.3 Davenport's bounds for G(5) and G(6)
6.4 Exercises
Vinogradov's upper bound for G(k)
7.1 Some remarks on Vinogradov's mean
value theorem
7.2 Preliminary estimates
7.3 An asymptotic formula for J(X)
7.4 Vinogradov's upper bound for G(k)
7.5 Exercises
A ternary additive problem
8.1 A general conjecture
8.2 Statement of the theorem
8.3 Definition of major and minor arcs
8.4 The treatment of n
8.5 The major arcs y(q.a)
8.6 The singular series
8.7 Completion of the proof of Theorem 8.
8.8 Exercises
Homogeneous equations and Birch's theorem
9.1 Introduction
9.2 Additive homogeneous equations
9.3 Birch's theorem
9.4 Exercises
A theorem of Roth
10.1 Introduction
10.2 Roth's theorem
10.3 A theorem of Furstenburg and Sarkozy
10.4 The definition of major and minor arcs
10.5 The contribution from the minor arcs
10.6 The contribution from the major arcs
10.7 Completion of the proof of Theorem 10.2
10.8 Exercises
Diophantine inequalities
11.1 A theorem of Davenport and Heilbronn
11.2 The definition of major and minor arcs
11.3 The treatment of the minor arcs
11.4 The major arc
11.5 Exercises
Wooley's upper bound for G(k)
12.1 Smooth numbers
12.2 The fundamental lemma
12.3 Successive efficient differences
12.4 A mean value theorem
12.5 Wooley's upper bound for G(k)
12.6 Exercises
Bibliography
Index
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