Format:
Online-Ressource (XI, 435 p. 4 illus, digital)
ISBN:
9781461438106
,
1280802820
,
9781280802829
Series Statement:
SpringerLink
Content:
Preface -- Introduction -- 1. Ranks and Cranks, Part I -- 2. Ranks and Cranks, Part II -- 3. Ranks and Cranks, Part III -- 4. Ramanujan's Unpublished Manuscript on the Partition and Tau Functions -- 5. Theorems about the Partition Function on Pages 189 and 182 -- 6. Congruences for Generalized Tau Functions on Page 178 -- 7. Ramanujan's Forty Identities for the Rogers-Ramanujan Functions -- 8. Circular Summation -- 9. Highly Composite Numbers -- Scratch Work -- Location Guide -- Provenance -- References.
Content:
In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge to examine the papers of the late G.N. Watson. Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan. This manuscript was soon designated, "Ramanujan's lost notebook." Its discovery has frequently been deemed the mathematical equivalent of finding Beethoven's tenth symphony. This volume is the third of five volumes that the authors plan to write on Ramanujan’s lost notebook and other manuscripts and fragments found in The Lost Notebook and Other Unpublished Papers, published by Narosa in 1988. The ordinary partition function p(n) is the focus of this third volume. In particular, ranks, cranks, and congruences for p(n) are in the spotlight. Other topics include the Ramanujan tau-function, the Rogers–Ramanujan functions, highly composite numbers, and sums of powers of theta functions. Review from the second volume: "Fans of Ramanujan's mathematics are sure to be delighted by this book. While some of the content is taken directly from published papers, most chapters contain new material and some previously published proofs have been improved. Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come. The next installment in this series is eagerly awaited." - MathSciNet Review from the first volume: "Andrews a nd Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society.
Note:
Description based upon print version of record
,
Ramanujan's Lost Notebook; Preface; Contents; 1 Introduction; 2 Ranks and Cranks, Part I; 2.1 Introduction; 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; 3.2 Preliminary Results; 3.3 The 2-Dissection for F(q); 3.4 The 3-Dissection for F(q); 3.5 The 5-Dissection for F(q); 3.6 The 7-Dissection for F(q); 3.7 The 11-Dissection for F(q); 3.8 Conclusion; 4 Ranks and Cranks, Part III; 4.1 Introduction; 4.2 Key Formulas on Page 59
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4.3 Proofs of Entries 4.2.1 and 4.2.34.4 Further Entries on Pages 58 and 59; 4.5 Congruences for the Coefficients lambdan on Pages 179 and 180; 4.6 Page 181: Partitions and Factorizations of Crank Coefficients; 4.7 Series on Pages 63 and 64 Related to Cranks; 4.8 Ranks and Cranks: Ramanujan's Influence Continues; 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 tau(n); 5.1 The Congruence p(5n+4)0(mod5)
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5.2 Divisibility of tau(n) by 55.3 The Congruence p(25n+24)0(mod25); 5.4 Congruences Modulo 5k; 5.5 Congruences Modulo 7; 5.6 Congruences Modulo 7, Continued; 5.7 Congruences Modulo 49; 5.8 Congruences Modulo 49, Continued; 5.9 The Congruence p(11n+6)0(mod11); 5.10 Congruences Modulo 11, Continued; 5.11 Divisibility by 2 or 3; 5.12 Divisibility of tau(n); 5.13 Congruences Modulo 13; 5.14 Congruences for p(n) Modulo 13; 5.15 Congruences to Further Prime Moduli; 5.16 Congruences for p(n) Modulo 17, 19, 23, 29, or 31; 5.17 Divisibility of tau(n) by 23; 5.18 The Congruence p(121n-5)0(mod121)
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5.19 Divisibility of tau(n) for Almost All Values of n5.20 The Congruence p(5n+4)0(mod5), Revisited; 5.21 The Congruence p(25n+24)0(mod25), Revisited; 5.22 Congruences for p(n) Modulo Higher Powers of 5; 5.23 Congruences for p(n) Modulo Higher Powers of 5, Continued; 5.24 The Congruence p(7n+5)0(mod7); 5.25 Commentary; 5.1 The Congruence p(5n+4)0(mod5); 5.2 Divisibility of tau(n) by 5; 5.4 Congruences Modulo 5k; 5.5 Congruences Modulo 7; 5.6 Congruences Modulo 7, Continued; 5.7 Congruences Modulo 49; 5.8 Congruences Modulo 49, Continued; 5.9 The Congruence p(11n+6)0(mod11)
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5.10 Congruences Modulo 11, Continued5.11 Divisibility by 2 or 3; 5.12 Divisibility of tau(n); 5.13 Congruences Modulo 13; 5.14 Congruences for p(n) Modulo 13; 5.15 Congruences to Further Prime Moduli; 5.16 Congruences for p(n) Modulo 17, 19, 23, 29, or 31; 5.17 Divisibility of tau(n) by 23; 5.18 The Congruence p(121n-5)0(mod121); 5.19 Divisibility of tau(n) for Almost All Values of n; 5.20 The Congruence p(5n+4)0(mod5), Revisited; 5.23 Congruences for p(n) Modulo Higher Powers of 5, Continued; 5.24 The Congruence p(7n+5)0(mod7); 6 Theorems about the Partition Function on Pages 189 and 182
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6.1 Introduction
Additional Edition:
ISBN 9781461438090
Additional Edition:
Buchausg. u.d.T. Andrews, George E., 1938 - Ramanujan's Lost Notebook ; 3 New York, NY : Springer, 2013 ISBN 9781461438090
Language:
English
Subjects:
Mathematics
DOI:
10.1007/978-1-4614-3810-6
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
(lizenzpflichtig)
Author information:
Berndt, Bruce C. 1939-
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