拉马努金遗失笔记（第三卷）

Introduction Ranks and Cranks, Part I

2.1 Introduction 9

2.2 Proof of Entry 2.1.1

2.3 Background for Entries 2.1.2 and 2.1.4

2.4 Proof of Entry 2.1.2

2.5 Proof of Entry 2.1.4

2.6 Proof of Entry 2.1.5

3 Ranks and Cranks, Part II

3.1 Introduction

32 Preliminary Results.……………

3.3 The 2-Dissection for F(a) .............

3.4 The 3-Dissection for F(q

3.5 The 5-Dissection for F(q)

3.6 The 7-Dissection for F(a

3.7 The 1l-Dissection for F(a

3.8 Conclusion

4 Ranks and Cranks. Part III

4.1 Introduction

4.2 Key Formulas on Page 59..'o3

4.3 Proofs of Entries 4.2.1 and 4.2.3

4.54 CongruencesFurtherEntriesfor theon PagesCoefficients58and An59 on Pages 179 and 180 74

4.6 Page 181: Partitions and Factorizations of Crank Coefficients. 82

4.7 Series on Pages 63 and 64 Related to Cranks

4.8 Ranks and Cranks: Ramanujan's Influence Continues..,..86

4.8.1 Congruences and Related Work

4.8.2 Asymptotics and Related Analysis

4.8.3 Combinatorics

4.8.4 Inequalities

4.8.5 Generalizations

5 Ramanujan,s Unpublished Manuscript on the Partition

and Tau Functions

5.0 Congruences for T(n)

5.1 The Congruence p(5n 4)=0(mod 5)

5.2 Divisibility of r(n) by 5

53 The Congruence p(257 24)≡0(mod25)………97

5.4 Congruences Modulo 5k

5.5 Congruences Modulo 7

5.6 Congruences Modulo 7, Continued

5.7 Congruences Modulo 49

58 Congruences Modulo49, Continued..………104

5.9 The Congruence p(1ln 6)=0(mod 11)

5.10 Congruences Modulo11, Continued..……………107

11 Divisibility by 2 or 3

5.12 Divisibility of T(n)

13 Congruences Modulo 13..................................119

5.14 Congruences for p(n) Modulo 13

5.15 Congruences to Further Prime Moduli........... 123 5.16 Congruences for p(n) Modulo 17, 19, 23, 29, or 31.......... 125 57 Divisibility of T(n)by23...……127 5. 18 The Congruence p(121n-5)=0(mod 121)................ 129 5. 19 Divisibility of T(n)for...

5.22 Congruences for p(n) Modulo Higher Powers of 5

5.23 Congruences for p(n) Modulo Higher Powers of 5, Continued. 136

5.24 The Congruence p(7n 5)=0(mod 7)

5.25 Commentary..,.… 1 The Congruence p(5n 4)=0(mod 5)

5.2 Divisibility of T (n) by 5

5.4 Congruences Modulo 5

5.5 Congruences Modulo 7............

56 Congruences Modulo7, Continued..……144

5.7 Congruences Modulo 49

58 Congruences Modulo49, Continued..……145

5.9 The Congruence p(117 6)≡0(mod11).………145

5.10 Congruences Modulo 11, Continued............ 146 11 Divisibility by 2 or 3 12 Divisibility of T(n)

5.13 Congruences Modulo 13

5.14 Congruences for p(n) Modulo 13

15 Congruences to Further Prime Moduli

5.16 Congruences for p(n) Modulo 17, 19, 23, 29, or 31 159

17 Divisibility of T(n) by 23 5.18 The Congruence p(12In-5)=0(mod 121)

5.19 Divisibility of T(n) for Almost All Values of n 177

5.20 The Congruence p(5n 4)=0(mod 5),Revisited 178

5.23 Congruences for p(n) Modulo Higher Powers of 5, Continued.179

5.24 The Congruence p(7n 5)=0(mod 7)

6 Theorems about the Partition Function on Pages 189 and

182 6.1 Introduction 6.2 The Identities for Modulus 5.............................. 183

6.3 The Identities for Modulus 7

6.4 Two Beautiful, False, but Correctable Claims of Ramanujan.193

6.5Page182.

6.6 Further Remarks

7 Congruences for Generalized Tau Functions on Page 178..205

7.1 Introduction

7.2 Proofs

8 Ramanujan's Forty Identities for the Rogers-Ramanujan

Functions

8.1 Introduction

8.2 Definitions and Preliminary Results

8.3 The Forty Identities

8.4 The Principal Ideas Behind the Proofs 229

8.5 Proofs of the 40 Entries 243

8.6 Other Identities for G(a) and H(g and Final Remarks...333

9 Circular Summation

1 Introduction............

9.2 Proof of Entry 9.1.1

9.3 Reformulations

9.4 Special Cases

10 Highly Composite Numbers

Scratch Work

Location Guide

Provenance

References

附录I拉马努金的中国知音:数学家刘治国的“西天取经”之旅附录II刘治国教授访谈

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