Hardy-Littlewood方法（第2版）（英文版）

Hardy-Littlewood方法（第2版）（英文版）

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作者: R.C.Vaughan 著

出版社: 世界图书出版公司

出版时间: 1998-08

版次: 1

ISBN: 9787506239226

定价: 35.00

装帧: 平装

开本: 其他

纸张: 胶版纸

页数: 232页

分类: 自然科学

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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. Contents

Preface

Preface to second edition

Notation

1 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

2 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

3 Goldbach's problems

　3.1 The ternary Goldbach problem

　3.2 The binary Goldbach problem

　3.3 Exercises

4 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

5 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

6 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

7 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

8 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

9 Homogeneous equations and Birch's theorem

……

10 A theorem of Roth

11 Diophantine inequalities

12 Wooley's upper bound for G(k)

Bibliography

Index
内容简介:
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.
目录:
Contents

Preface

Preface to second edition

Notation

1 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

2 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

3 Goldbach's problems

　3.1 The ternary Goldbach problem

　3.2 The binary Goldbach problem

　3.3 Exercises

4 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

5 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

6 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

7 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

8 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

9 Homogeneous equations and Birch's theorem

……

10 A theorem of Roth

11 Diophantine inequalities

12 Wooley's upper bound for G(k)

Bibliography

Index

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