PREFACE

Hardy in his thirties held the view that the late years of a mathematician's life were spent most profitably in writing books; I remember a particular conversation about this, and though we never spoke of the matter again it remained an understanding. The level below his best at which a man is prepared to go on working at full stretch is a matter of temperament; Hardy made his decision, and while of course he continued to publish papers his last years were mostly devoted to books; whatever has been lost, mathematical literature has greatly gained. All his books gave him some degree of pleasure, but this one, his last, was his favourite. When embarking on it he told me that he believed in its value (as he well might), and also that he looked forward to the task with enthusiasm. He had actually given lectures on the subject at intervals ever since his return to Cambridge in 1931, and had at one time or another lectured on everything in the book except Chapter XIII. The title holds curious echoes of the past, and of Hardy's past. Abel wrote in 1828: 'Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever.' In the ensuing period of critical revision they were simply rejected. Then came a time when it was found that something after all could be done about them. This is now a matter of course, but in the early years of the century the subject, while in no way mystical or unrigorous, was regarded as sensational, and about the present title, now colourless, there hung an aroma of paradox and audacity.

J. E. LITTLEWOODAugust 1948

前言

三十多岁的哈代认为,数学家的晚年大部分时间都花在写书上;我记得关于这件事的一次特别谈话,虽然我们再也没有谈过这件事,但它仍然是一种理解.如果一个人的工作始终低于他的最佳水平,他须得耐得住性子才能持续工作;哈代做出了自己的决定,当然,虽然他继续发表论文,但他的最后几年主要致力于写书;无论失去了什么,他都为数学文献作出了巨大的贡献.他所有的书都给了他一定程度的乐趣,这本是他的最后一本,却是他的最爱.开始写时,他告诉我他相信它的价值(尽管他很可能),而且他对这项任务充满热情.实际上,自从他 1931 年回到剑桥以来,他就每隔一段时间就这个主题进行演讲,并且时不时地对本书中的所有内容进行演讲,除了第十三章.标题包含对过去和哈代过去的奇怪回声.亚伯在 1828 年写道:"发散级数是魔鬼发明的,以它们作为基础的任何证明都是可耻的."在随后的批判性修订期间,他们只是被回避了.然后到了一个时候,我们发现还是可以用它们做些证明的.这个主题现在看来是理所当然的事情,然而在本世纪初,它虽然既不神秘又并非不严谨,但被认为是耸人听闻的,现今对它的偏见已经消失,它转而散发着悖论和大胆的气息.

约翰·恩瑟·李特尔伍德1948年八月

NOTE

Professor Hardy, who died on 1 December 1947, had sent the galleys of Chapters I-X to the press, and read the remaining galleys, before he felt unable to continue the work. Dr. H. G. Eggleston and I, who had also been reading the proofs, completed their revision in both galley and page form. Professor W. W. Rogosinski read the manuscript of Chapters I-II and XI-XII, and Miss S. M. Edmonds that of Chapter X; and I also read the book in manuscript. Dr. Eggleston checked all the references, drew up the lists of authors and definitions, and drafted the general index; and I added the note on conventions. My own task has been greatly lightened by Dr. Eggleston's help, and also by the care and consideration of the Clarendon Press.

L. S. BOSANQUETAugust 1948

哈代教授于 1947 年 12 月 1 日去世,在他感到无法继续工作之前,他已将第一章至第十章的校样寄给了出版社,并阅读了其余的校样. H. G. Eggleston 博士和我也一直在阅读校样,他们完成了校样和版式的修订. W. W. Rogosinski 教授阅读了第一至第二章和第十一至第十二章的手稿,S. M. Edmonds 小姐阅读了第十章的手稿;我也阅读了这本书的手稿. Eggleston 博士检查了所有参考文献,起草了作者列表,定义列表和总索引;我添加了关于约定的注释. Eggleston 博士的帮助以及克拉伦登出版社的关心极大地减轻了我的任务.

兰斯洛特·斯蒂芬·博桑奎特1948年八月

CONTENTS

Ⅰ. NOTE ON CONVENTIONS

1.1. The sum of a series

1.2. Some calculations with divergent series

1.3. First definitions

1.4. Regularity of a method

1.5. Divergent integrals and generalized limits of a function of a continuous variable

1.6. Some historical remarks

1.7. A note on the British analysts of the early nineteenth century

NOTES ON CHAPTER Ⅰ

Ⅱ. SOME HISTORICAL EXAMPLES

2.1. Introduction

A.

2.2. The functional equations for $\zeta(s),\eta(s)$ and $L(s)$

2.3. Euler's verification

B.

2.4. Summation of the series

2.5. The asymptotic nature of the series

2.6. Numerical computations

C.

2.7. Fourier's theorem

2.8. Fourier's first formula for the coefficients

2.9. Other forms of the coefficients and the series

2.10. The validity of Fourier's formulae

D.

2.11. Heaviside on divergent series

2.12. The generalized exponential series

2.13. The series $\sum\phi^{(r)}(x)$

2.14. The generalized binomial series

NOTES ON CHAPTER Ⅱ

Ⅲ. GENERAL THEOREMS

3.1. Generalities concerning linear transformations

3.2. Regular transformations

3.3. Proof of Theorems 1 and 2

3.4. Proof of Theorem 3

3.5. Variants and analogues

3.6. Positive transformations

3.7. Knopp's kernel theorem

3.8. An application of Theorem 2

3.9. Dilution of series

NOTES ON CHAPTER Ⅲ

Ⅳ. SPECIAL METHODS OF SUMMATION

4.1. Nörlund means

4.2. Regularity and consistency of Nörlund means

4.3. Inclusion

4.4. Equivalence

4.5. Another theorem concerning inclusion

4.6. Euler means

4.7. Abelian means

4.8. A theorem of inclusion for Abelian means

4.9. Complex methods

4.10. Summability of $1-1+1-\cdots$ by special Abelian methods

4.11. Lindelöf's and Mittag-Leffler's methods

4.12. Means defined by integral functions

4.13. Moment constant methods

4.14. A theorem of consistency

4.15. Methods ineffective for the series $1-1+1-\cdots$

4.16. Riesz's typical means

4.17. Methods suggested by the theory of Fourier series

4.18. A general principle

NOTES ON CHAPTER Ⅳ

Ⅴ. ARITHMETIC MEANS (1)

5.1. Introduction

5.2. Hölder's means

5.3. Simple theorems concerning Hölder summability

5.4. Cesàro means

5.5. Means of non-integral order

5.6. A theorem concerning integral resultants

5.7. Simple theorems concerning Cesàro summability

5.8. The equivalence theorem

5.9. Mercer's theorem and Schur's proof of the equivalence

5.10. Other proofs of Mercer's theorem

5.11. Infinite limits

5.12. Cesàro and Abel summability .

5.13. Cesàro means as Nörlund means

5.14. Integrals

5.15. Theorems concerning summable integrals

5.16. Riesz's arithmetic means

5.17. Uniformly distributed sequences

5.18. The uniform distribution of $\{n^2a\}$

NOTES ON CHAPTER Ⅴ

Ⅵ. ARITHMETIC MEANS (2)

6.1. Tauberian theorems for Cesàro summability

6.2. Slowly oscillating and slowly decreasing functions

6.3. Another Tauberian condition

6.4. Convexity theorems

6.5. Convergence factors

6.6. The factor $(n+1)^{-s}$

6.7. Another condition for summability

6.8. Integrals

6.9. The binomial series

6.10. The series $\sum n^\alpha e^{ni\theta}$

6.11. The case $\beta=-1$

6.12. The series $\sum n^{-b}e^{A\ln a}$

NOTES ON CHAPTER Ⅵ

Ⅶ. TAUBERIAN THEOREMS FOR POWER SERIES

7.1. Abelian and Tauberian theorems

7.2. Tauber's first theorem

7.3. Tauber's second theorem

7.4. Applications to general Dirichlet's series

7.5. The deeper Tauberian theorems

7.6. Proof of Theorems 96 and 96 a

7.7. Proof of Theorems 91 and 91 a

7.8. Further remarks on the relations between the theorems of § 7.5

7.9. The series $\sum n^{-1-ic}$

7.10. Slowly oscillating and slowly decreasing functions

7.11. Another generalization of Theorem 98

7.12. The method of Hardy and Littlewood

7.13. The “high indices” theorem

NOTES ON CHAPTER Ⅶ

Ⅷ. THE METHODS OF EULER AND BOREL (1)

8.1. Introduction

8.2. The $(E, q)$ method

8.3. Simple properties of the $(E, q)$ method

8.4. The formal relations between Euler's and Borel's methods

8.5. Borel's methods

8.6. Normal, absolute, and regular summability

8.7. Abelian theorems for Borel summability

8.8. Analytic continuation of a function regular at the origin: the polygon of summability

8.9. Series representing functions with a singular point at the origin

8.10. Analytic continuation by other methods

8.11. The summability of certain asymptotic series

NOTES ON CHAPTER VIII

Ⅸ. THE METHODS OF EULER AND BOREL (2)

9.1. Some elementary lemmas

9.2. Proof of Theorem 137

9.3. Proof of Theorem 139

9.4. Another elementary lemma

9.5. Ostrowski's theorem on over-convergence

9.6. Tauberian theorems for Borel summability

9.7. Tauberian theorems (

9.8. Examples of series not summable (B)

9.9. A theorem in the opposite direction

9.10. The $(e, c)$ method of summation

9.11. The circle method of summation

9.12. Further remarks on Theorems 150-5

9.13. The principal Tauberian theorem

9.14. Generalizations

9.15. The series $\sum z^n$

9.16. Valiron's methods

NOTES ON CHAPTER IX

Ⅹ. MULTIPLICATION OF SERIES

10.1. Formal rules for multiplication

10.2. The classical theorems for multiplication by Cauchy's rule

10.3. Multiplication of summable series

10.4. Another theorem concerning convergence

10.5. Further applications of Theorem 170

10.6. Alternating series

10.7. Formal multiplication

10.8. Multiplication of integrals

10.9. Euler summability

10.10. Borel summability

10.11. Dirichlet multiplication

10.12. Series infinite in both directions

10.13. The analogues of Cauchy's and Mertens's theorems

10.14. Further theorems

10.15. The analogue of Abel's theorem

NOTES ON CHAPTER Ⅹ

Ⅺ. HAUSDORFF MEANS

11.1. The transformation $\delta$

11.2. Expression of the $(E, q)$ and $(C, 1)$ transformations in terms of $\delta$

11.3. Hausdorff's general transformation

11.4. The general Hölder and Cesàro transformations as $\mathfrak H$ transformations

11.5. Conditions for the regularity of a real Hausdorff transformation

11.6. Totally monotone sequences

11.7. Final form of the conditions for regularity

11.8. Moment constants

11.9. Hausdorff's theorem

11.10. Inclusion and equivalence of $\mathfrak H$ methods

11.11. Mercer's theorem and the equivalence theorem for Hölder and Cesàro means

11.12. Some special cases

11.13. Logarithmic cases

11.14. Exponential cases

11.15. The Legendre series for $\chi(x)$

11.16. The moment constants of functions of particular classes

11.17. An inequality for Hausdorff means

11.18. Continuous transformations

11.19. Quasi-Hausdorff transformations

11.20. Regularity of a quasi-Hausdorff transformation

11.21. Examples

NOTES ON CHAPTER Ⅺ

Ⅻ. WIENER'S TAUBERIAN THEOREMS

12.1. Introduction

12.2. Wiener's condition

12.3. Lemmas concerning Fourier transforms

12.4. Lemmas concerning the class $U$

12.5. Final lemmas

12.6. Proof of Theorems 221 and 220

12.7. Wiener's second theorem

12.8. Theorems for the interval $(0,\infty)$

12.9. Some special kernels

12.10. Application of the general theorems to

12.11. Applications to the theory of primes

12.12. One-sided conditions

12.13. Vijayaraghavan's theorem

12.14. Proof of Theorem 238

12.15. Borel summability

12.16. Summability $(R, 2)$

NOTES ON CHAPTER Ⅻ

ⅩⅢ. THE EULER-MACLAURIN SUM FORMULA

13.1. Introduction

13.2. The Bernoullian numbers and functions

13.3. The associated periodic functions

13.4. The signs of the functions $\phi_n(x)$

13.5. The Euler-Maclaurin sum formula

13.6. Limits as $n\to\infty$

13.7. The sign and magnitude of the remainder term

13.8. Poisson's proof of tho Euler-Maclaurin formula

13.9. A formula of Fourier

13.10. The case $f(x)=x^{-s}$ and the Riemann zeta-function

13.11. The case $f(x)=log(x+c)$ and Stirling's theorem

13.12. Generalization of the formulae

13.13. Other formulae for $C$

13.14. Investigation of the Euler-Maclaurin formula by complex integration

13.15. Summability of the Euler-Maclaurin series

13.16. Additional remarks

13.17. The $\mathfrak R$ definition of the sum of a divergent series

NOTES ON CHAPTER ⅩⅢ

APPENDIX Ⅰ. On the evaluation of certain definite integrals by means of divergent series

APPENDIX Ⅱ. The Fourier kernels of certain methods of summation

APPENDIX Ⅲ. On Rdsmann and Abel summability

APPENDIX Ⅳ. On Lambert and Ingham summability

APPENDIX Ⅴ. Two theorems of M. L. Cartwright

LIST OF BOOKS

LIST OF PERIODICALS

LIST OF AUTHORS

LIST OF DEFINITIONS

GENERAL INDEX

评论

来自stackexchange:

Ah, yes, a book such as this is far away from undergrad curriculum, and even from basic grad-level math curriculum. It is not playing according to some quasi-orthodox set of rules, axioms, whatever one might say...

And, in my opinion, all the more charming and interesting because it is about "live" mathematics.

No, basic as-in-school ideas from calculus are wildly insufficient. Literally, "calculus" seems to teach us that "non-convergent " is

The naive form of "summing divergent series" seems to ask to capriciously assign values to "infinite sums" that don't have an "easy" (=convergent) sense. But, naturally, there are external constraints that prevent us from being completely capricious. However, in Hardy's time, there was more-limited vocabulary and framework for expressing "external constraints". Instead, by accident, his descriptive vocabulary would sound to us as "definitions"... specifically, lacking motivation or description of interaction/compatibility with "external requirements".

In particular, he implicity, perhaps subliminally, and certainly rarely overtly in that text, there are implicit requirements of compatibility with meromorphic continuations (which Euler already understood at least subliminally), and other not-just-calculus issues.

So, again, no, basic calculus is not the context in which to understand that book. The standard curriculum (such as it is... sigh...) wouldn't provide the relevant formal (sigh...) background until second or third year of math-grad-school, if then. Small wonder it's hard to understand directly/naively.

Yet, yes, the book does have charms, and does not consciously try to oppress the reader...