Electronic ISBN:  9781470467128 
Product Code:  CHEL/159.E 
List Price:  $58.00 
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Book DetailsAMS Chelsea PublishingVolume: 159; 1927; 355 pp
The influence of Ramanujan on number theory is without parallel in mathematics. His papers, problems, and letters have spawned a remarkable number of later results by many different mathematicians. Here, his 37 published papers, most of his first two and last letters to Hardy, the famous 58 problems submitted to the Journal of the Indian Mathematical Society, and the commentary of the original editors (Hardy, Seshu Aiyar and Wilson) are reprinted again, after having been unavailable for some time.
In this printing of Ramanujan's collected papers, Bruce Berndt provides an annotated guide to Ramanujan's work and to the mathematics it inspired over the last threequarters of a century. The historical development of ideas is traced in the commentary and by citations to the copious references. The editor has done the mathematical world a tremendous service that few others would be qualified to do.ReadershipGraduate students and research mathematicians interested in number theory.

Table of Contents

Front Cover

CONTENTS

PREFACE

SRINIVASA RAMANUJAN (18871920) BY P. V. SESHU AIYAR AND R. RAMACHANDRA RAO

SRINIVASA RAMANUJAN (18871920) BY G. H. HARDY

Preface to the Third Printing

1 SOME PROPERTIES OF BERNOULLI'S NUMBERS

2 ON QUESTION 330 OF PROFESSOR SANJANA

3 NOTE ON A SET OF SIMULTANEOUS EQUATIONS

4 IRREGULAR NUMBERS

5 SQUARING THE CIRCLE

6 MODULAR EQUATIONS AND APPROXIMATIONS TO π

7 ON THE INTEGRAL ∫ tan^(1)t/t dt

8 ON THE NUMBER OF DIVISORS OF A NUMBER

9 ON THE SUM OF THE SQUARE ROOTS OF THE FIRST n NATURAL NUMBERS

10 ON THE PRODUCT ∏[1 + (x/a+nd)^3]

11 SOME DEFINITE INTEGRALS

12 SOME DEFINITE INTEGRALS CONNECTED WITH GAUSS'S SUMS

13 SUMMATION OF A CERTAIN SERIES

14 NEW EXPRESSIONS FOR RIEMANN'S FUNCTIONS ξ(s) AND Ξ(t)

15 HIGHLY COMPOSITE NUMBERS

16 ON CERTAIN INFINITE SERIES

17 SOME FORMULÆ IN THE ANALYTIC THEORY OF NUMBERS

18 ON CERTAIN ARITHMETICAL FUNCTIONS

19 A SERIES FOR EULER'S CONSTANT y

20 ON THE EXPRESSION OF A NUMBER IN THE FORM ax^2 + by^2 + cz62 + du^2

21 ON CERTAIN TRIGONOMETRICAL SUMS AND THEIR APPLICATIONS IN THE THEORY OF NUMBERS

22 SOME DEFINITE INTEGRALS

23 SOME DEFINITE INTEGRALS

24 A PROOF OF BERTRAND'S POSTULATE

25 SOME PROPERTIES OF p(n), THE NUMBER OF PARTITIONS OF n *

26 PROOF OF CERTAIN IDENTITIES IN COMBINATORY ANALYSIS

27 A CLASS OF DEFINITE INTEGRALS

28 CONGRUENCE PROPERTIES OF PARTITIONS

29 ALGEBRAIC RELATIONS BETWEEN CERTAIN INFINITE PRODUCTS

30 CONGRUENCE PROPERTIES OF PARTITIONS

31 UNE FORMULE ASYMPTOTIQUE POUR LE NOMBRE DES PARTITIONS DE n

32 PROOF THAT ALMOST ALL NUMBERS n ARE COMPOSED OF ABOUT log log n PRIME FACTORS

33 ASYMPTOTIC FORMULÆ IN COMBINATORY ANALYSIS

34 ASYMPTOTIC FORMULÆ FOR THE DISTRIBUTION OF INTEGERS OF VARIOUS TYPES

35 THE NORMAL NUMBER OF PRIME FACTORS OF A NUMBER n

36 ASYMPTOTIC FORMULÆ IN COMBINATORY ANALYSIS

37 ON THE COEFFICIENTS IN THE EXPANSIONS OF CERTAIN MODULAR FUNCTIONS

QUESTIONS AND SOLUTIONS

APPENDIX I: NOTES ON THE PAPERS

APPENDIX II. FURTHER EXTRACTS FROM RAMANUJAN'S LETTERS TO G. H. HARDY

COMMENTARY ON RAMANUJAN'SCOLLECTED PAPERS

Back Cover


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The influence of Ramanujan on number theory is without parallel in mathematics. His papers, problems, and letters have spawned a remarkable number of later results by many different mathematicians. Here, his 37 published papers, most of his first two and last letters to Hardy, the famous 58 problems submitted to the Journal of the Indian Mathematical Society, and the commentary of the original editors (Hardy, Seshu Aiyar and Wilson) are reprinted again, after having been unavailable for some time.
In this printing of Ramanujan's collected papers, Bruce Berndt provides an annotated guide to Ramanujan's work and to the mathematics it inspired over the last threequarters of a century. The historical development of ideas is traced in the commentary and by citations to the copious references. The editor has done the mathematical world a tremendous service that few others would be qualified to do.
Graduate students and research mathematicians interested in number theory.

Front Cover

CONTENTS

PREFACE

SRINIVASA RAMANUJAN (18871920) BY P. V. SESHU AIYAR AND R. RAMACHANDRA RAO

SRINIVASA RAMANUJAN (18871920) BY G. H. HARDY

Preface to the Third Printing

1 SOME PROPERTIES OF BERNOULLI'S NUMBERS

2 ON QUESTION 330 OF PROFESSOR SANJANA

3 NOTE ON A SET OF SIMULTANEOUS EQUATIONS

4 IRREGULAR NUMBERS

5 SQUARING THE CIRCLE

6 MODULAR EQUATIONS AND APPROXIMATIONS TO π

7 ON THE INTEGRAL ∫ tan^(1)t/t dt

8 ON THE NUMBER OF DIVISORS OF A NUMBER

9 ON THE SUM OF THE SQUARE ROOTS OF THE FIRST n NATURAL NUMBERS

10 ON THE PRODUCT ∏[1 + (x/a+nd)^3]

11 SOME DEFINITE INTEGRALS

12 SOME DEFINITE INTEGRALS CONNECTED WITH GAUSS'S SUMS

13 SUMMATION OF A CERTAIN SERIES

14 NEW EXPRESSIONS FOR RIEMANN'S FUNCTIONS ξ(s) AND Ξ(t)

15 HIGHLY COMPOSITE NUMBERS

16 ON CERTAIN INFINITE SERIES

17 SOME FORMULÆ IN THE ANALYTIC THEORY OF NUMBERS

18 ON CERTAIN ARITHMETICAL FUNCTIONS

19 A SERIES FOR EULER'S CONSTANT y

20 ON THE EXPRESSION OF A NUMBER IN THE FORM ax^2 + by^2 + cz62 + du^2

21 ON CERTAIN TRIGONOMETRICAL SUMS AND THEIR APPLICATIONS IN THE THEORY OF NUMBERS

22 SOME DEFINITE INTEGRALS

23 SOME DEFINITE INTEGRALS

24 A PROOF OF BERTRAND'S POSTULATE

25 SOME PROPERTIES OF p(n), THE NUMBER OF PARTITIONS OF n *

26 PROOF OF CERTAIN IDENTITIES IN COMBINATORY ANALYSIS

27 A CLASS OF DEFINITE INTEGRALS

28 CONGRUENCE PROPERTIES OF PARTITIONS

29 ALGEBRAIC RELATIONS BETWEEN CERTAIN INFINITE PRODUCTS

30 CONGRUENCE PROPERTIES OF PARTITIONS

31 UNE FORMULE ASYMPTOTIQUE POUR LE NOMBRE DES PARTITIONS DE n

32 PROOF THAT ALMOST ALL NUMBERS n ARE COMPOSED OF ABOUT log log n PRIME FACTORS

33 ASYMPTOTIC FORMULÆ IN COMBINATORY ANALYSIS

34 ASYMPTOTIC FORMULÆ FOR THE DISTRIBUTION OF INTEGERS OF VARIOUS TYPES

35 THE NORMAL NUMBER OF PRIME FACTORS OF A NUMBER n

36 ASYMPTOTIC FORMULÆ IN COMBINATORY ANALYSIS

37 ON THE COEFFICIENTS IN THE EXPANSIONS OF CERTAIN MODULAR FUNCTIONS

QUESTIONS AND SOLUTIONS

APPENDIX I: NOTES ON THE PAPERS

APPENDIX II. FURTHER EXTRACTS FROM RAMANUJAN'S LETTERS TO G. H. HARDY

COMMENTARY ON RAMANUJAN'SCOLLECTED PAPERS

Back Cover