Hardy-Littlewood 方法（第2版）

     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