Ramanujan’s Lost Notebook George E. Andrews r Bruce C. Berndt Ramanujan’s Lost Notebook Part III George E. Andrews Bruce C. Berndt Department of Mathematics Department of Mathematics The Pennsylvania State University University of Illinois at University Park, PA Urbana–Champaign USA Urbana, IL USA ISBN 978-1-4614-3809-0 ISBN 978-1-4614-3810-6 (eBook) DOI 10.1007/978-1-4614-3810-6 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2005923547 Mathematics Subject Classification (2010): 11P82, 11P81, 11P84, 11P83, 33Dxx © Springer Science+Business Media New York 2012 This work is subject to copyright. 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Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) The CEO’s of Ranks and Cranks Freeman Dyson Frank Garvan Oliver Atkin H.P.F. Swinnerton-Dyer I felt the joy of an explorer who suddenly discov- ers the key to the language lying hidden in the hieroglyphs which are beautiful in themselves. –Rabindranath Tagore, The Religion of Man Preface This is the third of five volumes that the authors plan to write in their exami- nation of all the claims made by S. Ramanujan in The Lost Notebook and Other Unpublished Papers, which was published by Narosa in 1988. This publication contains the “Lost Notebook,” which was discovered by the first author in the spring of 1976 at the library of Trinity College, Cambridge. Also included therein are other partial manuscripts, fragments, and letters that Ramanujan wrote to G.H. Hardy from nursing homes during 1917–1919. Our third volume contains ten chapters and focuses on some of the most important and influen- tial material in The Lost Notebook and Other Unpublished Papers. At center stage is the partition function p(n). In particular, three chapters are devoted to ranks and cranks of partitions. Ramanujan’s handwritten manuscript on the partition and tau functions is also examined. vii Contents 1 Introduction ............................................... 1 2 Ranks and Cranks, Part I ................................. 9 2.1 Introduction............................................ 9 2.2 Proof of Entry 2.1.1 ..................................... 17 2.3 Background for Entries 2.1.2 and 2.1.4 ..................... 18 2.4 Proof of Entry 2.1.2 ..................................... 24 2.5 Proof of Entry 2.1.4 ..................................... 32 2.6 Proof of Entry 2.1.5 ..................................... 33 3 Ranks and Cranks, Part II ................................ 45 3.1 Introduction............................................ 45 3.2 PreliminaryResults...................................... 47 3.3 The 2-Dissection for F (q)................................. 48 3.4 The 3-Dissection for F (q)................................. 51 3.5 The 5-Dissection for F (q)................................. 56 3.6 The 7-Dissection for F (q)................................. 60 3.7 The 11-Dissection for F (q)................................ 63 3.8 Conclusion ............................................. 69 4 Ranks and Cranks, Part III ................................ 71 4.1 Introduction............................................ 71 4.2 KeyFormulasonPage59................................. 71 4.3 Proofs of Entries 4.2.1 and 4.2.3 ........................... 72 4.4 Further Entries on Pages 58 and 59 . ..................... 74 4.5 Congruences for the Coefficients λn on Pages 179 and 180 . 75 4.6 Page 181: Partitions and Factorizations of Crank Coefficients . 82 4.7 Series on Pages 63 and 64 Related to Cranks . .......... 84 4.8 Ranks and Cranks: Ramanujan’s Influence Continues . ...... 86 4.8.1 Congruences and Related Work ..................... 87 ix xContents 4.8.2 Asymptotics and Related Analysis . ............. 87 4.8.3 Combinatorics . ................................ 88 4.8.4 Inequalities....................................... 88 4.8.5 Generalizations . ................................ 88 5 Ramanujan’s Unpublished Manuscript on the Partition and Tau Functions ......................................... 89 5.0 Congruences for τ(n) .................................... 92 5.1 The Congruence p(5n +4)≡ 0(mod5)..................... 94 5.2 Divisibility of τ(n)by5 .................................. 95 5.3 The Congruence p(25n + 24) ≡ 0(mod25).................. 97 5.4 Congruences Modulo 5k .................................. 98 5.5 CongruencesModulo7................................... 99 5.6 CongruencesModulo7,Continued.........................100 5.7 CongruencesModulo49..................................102 5.8 CongruencesModulo49,Continued........................104 5.9 The Congruence p(11n +6)≡ 0(mod11)...................106 5.10CongruencesModulo11,Continued........................107 5.11 Divisibility by 2 or 3 . ................................110 5.12 Divisibility of τ(n).......................................113 5.13CongruencesModulo13..................................119 5.14 Congruences for p(n)Modulo13 ..........................121 5.15CongruencestoFurtherPrimeModuli .....................123 5.16 Congruences for p(n)Modulo17,19,23,29,or31...........125 5.17 Divisibility of τ(n)by23.................................127 5.18 The Congruence p(121n − 5) ≡ 0 (mod 121) . .............129 5.19 Divisibility of τ(n) for Almost All Values of n ...............130 5.20 The Congruence p(5n +4)≡ 0(mod5),Revisited............132 5.21 The Congruence p(25n + 24) ≡ 0 (mod 25), Revisited .........134 5.22 Congruences for p(n)ModuloHigherPowersof5............135 5.23 Congruences for p(n) Modulo Higher Powers of 5, Continued . 136 5.24 The Congruence p(7n +5)≡ 0(mod7).....................138 5.25Commentary............................................139 5.1 The Congruence p(5n +4)≡ 0(mod5).....................140 5.2 Divisibility of τ(n)by5 ..................................141 5.4 Congruences Modulo 5k ..................................143 5.5 CongruencesModulo7...................................144 5.6 CongruencesModulo7,Continued.........................144 5.7 CongruencesModulo49..................................144 5.8 CongruencesModulo49,Continued........................145 5.9 The Congruence p(11n +6)≡ 0(mod11)...................145 5.10CongruencesModulo11,Continued........................146 5.11 Divisibility by 2 or 3 . ................................147 5.12 Divisibility of τ(n).......................................150 5.13CongruencesModulo13..................................151 Contents xi 5.14 Congruences for p(n)Modulo13 ..........................153 5.15CongruencestoFurtherPrimeModuli .....................153 5.16 Congruences for p(n)Modulo17,19,23,29,or31...........159 5.17 Divisibility of τ(n)by23.................................176 5.18 The Congruence p(121n − 5) ≡ 0 (mod 121) .................177 5.19 Divisibility of τ(n) for Almost All Values of n ...............177 5.20 The Congruence p(5n +4)≡ 0(mod5),Revisited............178 5.23 Congruences for p(n) Modulo Higher Powers of 5, Continued . 179 5.24 The Congruence p(7n +5)≡ 0(mod7).....................179 6 Theorems about the Partition Function on Pages 189 and 182 ........................................................181 6.1 Introduction............................................181 6.2 TheIdentitiesforModulus5..............................183 6.3 TheIdentitiesforModulus7..............................186 6.4 Two Beautiful, False, but Correctable Claims of Ramanujan . 193 6.5 Page182...............................................195 6.6 FurtherRemarks........................................199 7 Congruences for Generalized Tau Functions on Page 178 ..205 7.1 Introduction............................................205 7.2 Proofs .................................................207 8 Ramanujan’s Forty Identities for the Rogers–Ramanujan Functions ..................................................217 8.1 Introduction............................................217