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Scratch Work Scratch Work Introduction In the introduction, partially on the basis of several pages of scratch work, we tendered the conjecture that Ramanujan had devoted his last hours to cranks before his suffering became too intense to work on mathematics in the last four days of his brief life. In this short appendix we briefly examine some of the pages of scratch work. Most of the scratch work that we can identify pertains to calculations involving theta functions, cranks, or the partition function. We emphasize that these pages contain no exciting discoveries. The scratch work gives us glimpses of some of Ramanujan’s thoughts, but perhaps more importantly, the scratch work demonstrates the importance of calculations for Ramanujan. We discuss pages in the order in which they appear in [283]. Page 61 As discussed in Chapter 4, the first five tables are preliminary versions of the tables given on page 179. These tables are followed by seven lists of num- bers, three belonging to residue classes 1 modulo 3, three belonging to residue classes −1 modulo 3, and the last belonging to multiples of 3. However, gen- erally, neither the numbers nor the values of p(n) for these n belong to the requisite residue classes. The seven classes contain a total of 71 numbers, with the largest being 130. Below the tables is the quotient of (apparently) infinite products (q; q)2 ∞ . (q3; q3)∞ The scratch work at the bottom of the page contains several theta functions depicted by the first several terms of their infinite series representations, in particular, G.E. Andrews, B.C. Berndt, Ramanujan’s Lost Notebook: Part III, 403 DOI 10.1007/978-1-4614-3810-6, c Springer Science+Business Media New York 2012 404 Scratch Work 1 − q − q2 + q5 + q7 − q12 − q15 −···= f(−q), 1 − q2 − q7 + q13 + q23 − q33 − q48 + ···= f(−q2, −q7), 1 − q4 − q5 + q17 + q19 − q39 − q42 + ···= f(−q4, −q5). We also see 1 − q7 1+q2 + q4 + q6 − q7 + q8 − q9 + q10 + ···= . 1 − q2 Page 65=73, 66 Inexplicably, the publisher photocopied page 65 twice. Calculations on these pages appear to be related to cranks. Page 72 Some of the calculations appear to be related to cranks. Pages 74–77 These pages may be related to the generating functions in Chapters 2 and 3. In the upper left-hand corner of page 76, we find the expressions 1 − q2 − q3 + ··· (1 − q5)(1 − q10) , , (1 − q)2(1 − q4)2 (1 − q)(1 − q4) (1 − q5)(1 − q10) 1 − q − q4 + ··· , . (1 − q2)(1 − q5) (1 − q2)2(1 − q3)2 Page 78 Printed upside down, we find (1 − q7 − q8 + q29 + q31 −···)(1 − q5 − q10 + q25 + q35 −···) = f(−q7, −q8)f(−q5, −q10). Immediately following, we find 1 − q − q4 + q7 + q13 − q18 − q27 + q34 + q46 −···= f(−q, −q4). We remark that we have inserted −··· in each instance above. Page 82 405 Page 79 If the series 1 − q + q2 + q6 + q8 − q9 + q10 − q11 +2q12 were extended to infinity, it would equal 10 9 10 10 10 5 9 (q; q )∞(q ; q )∞(q ; q )∞ f(−q )f(−q, −q ) = . (q2; q5)∞(q3; q5)∞ f(−q2, −q3) In the middle of the page, we see 1+q + q2 + q3 +2q4 + q6 + q7 +2q8 + q9, which, if the summands were extended to infinity, would be equal to (q5; q5)2 f 3(−q5) ∞ = . (q; q5)∞(q4; q5)∞ f(−q, −q4) Page 80 This page appears to be related to calculations related to cranks. Reading sideways, we again see many functions identical to or similar to others that we have seen before, including 2 3 5 5 5 5 f(−q , −q ) (q ; q )∞ (q ; q )∞ 5 2 4 5 2 , 5 4 5 , 2 5 3 5 . (q; q )∞(q ; q )∞ (q; q )∞(q ; q )∞ (q ; q )∞(q ; q )∞ Page 81 This page may be related to Chapters 2 and 3. Page 82 At the top of the page we see 1+q − q2 − q5 + q7 + q12 − q15 − q22 + q26 + q35 − q40 − ... 0 ∞ = (−1)nqn(3n−1)/2 − (−1)nqn(3n−1)/2, n=−∞ n=1 which is a false theta function. 406 Scratch Work Pages 83–85 These pages are likely related to calculations involving cranks. Pages 86–89 These pages contain a table of the residue classes of p(n)forn from 1 to nearly 200. The first column is n, the second is p(n) (mod 2), the third is p(n) (mod 3), the fourth is p(n) (mod 5), the fifth is p(n) (mod 7), and the sixth is p(n) (mod 11). For the first 17 values of n, the residue of p(n) modulo 13, 17, 19, and 23 are also given. On the right side of page 87, Ramanujan also tabulates the number of values in each residue class modulo 2, 3, 5, and 7 for the first and second 50 values of n. On page 86, Ramanujan appears to have calculated residues modulo various n for some arithmetic function that we cannot identify. Page 213 This page contains a small amount of scratch work on Eisenstein series, but no theorem is offered. Location Guide For each page of Ramanujan’s lost notebook on which we have discussed or proved entries in this book, we provide below a list of those chapters or entries in which these pages are discussed. Page 18 Entries 2.1.1, 3.5.1 Page 19 Entries 2.1.4, 2.1.5, 3.6.1 Page 20 Entries 2.1.1, 2.1.2, 3.5.1 Page 54 Entries 9.1.1, 9.4.1, 9.4.2, 9.4.3, 9.4.4, 9.4.5 Page 58 Section 4.4 Page 59 Entries 3.1.1, 4.2.1, 4.2.3, Section 4.4 Page 61 Section 4.5, Scratch Work G.E. Andrews, B.C. Berndt, Ramanujan’s Lost Notebook: Part III, 407 DOI 10.1007/978-1-4614-3810-6, c Springer Science+Business Media New York 2012 408 Location Guide Page 63 Section 4.7 Page 64 Section 4.7 Page 65 Scratch Work Page 66 Scratch Work Page 70 Entry 3.7.1 Page 72–89 Scratch Work Pages 135–177 Chapter 5 Page 178 Chapter 7 Page 179 Entries 3.3.1, 3.4.1, Section 4.5 Page 180 Section 4.5 Page 181 Section 4.6 Page 182 Entries 6.5.1, 6.5.2, 6.5.3 Page 184 Entry 2.1.3 Location Guide 409 Page 189 Entry 6.2.1,Entry6.2.2,Entry6.3.1,Entry6.3.2, Entry 6.4.1,Entry6.4.2,Entry6.4.3 Page 207 Section 6.6 Page 208 Section 6.6 Page 213 Scratch Work Pages 236–237 Chapter 8 Pages 238–243 Chapter 5 Page 248 Section 6.6 Page 252 Section 6.6 Pages 280–312 Chapter 10 Page 326 Entry 6.6.1, Section 6.6 Page 331 Entries 6.6.2–6.6.5 Page 333 Section 6.6 Provenance Chapter 2 A.O.L. Atkin and H.P.F. Swinnerton-Dyer, [28] F.G. Garvan, [144], [146] Chapter 3 B.C. Berndt, H.H. Chan, S.H. Chan, and W.-C. Liaw, [62] Chapter 4 B.C. Berndt, H.H. Chan, S.H. Chan, and W.-C. Liaw, [63], [64] W.-C. Liaw, [216] Chapter 5 B.C. Berndt and K. Ono, [67] Chapter 6 B.C. Berndt, A.J. Yee, and J. Yi, [70] B.C. Berndt, C. Gugg, and S. Kim, [66] Chapter 7 J.M. Rushforth, [305], [306] Chapter 8 B.C. Berndt, G. Choi, Y.-S. Choi, H. Hahn, B. P. Yeap, A. J. Yee, H. Yesilyurt, and J. Yi, [65] B.C. Berndt and H. Yesilyurt, [73] H. Yesilyurt, [347], [348] G.E. Andrews, B.C. Berndt, Ramanujan’s Lost Notebook: Part III, 411 DOI 10.1007/978-1-4614-3810-6, c Springer Science+Business Media New York 2012 412 Provenance Chapter 9 A. Berkovich, F.G. Garvan, and H. Yesilyurt, [53] H.H. Chan, Z.-G. Liu, and S.T. Ng, [101] S.H. Son, [320] P. Xu, [345] Chapter 10 J.-L. Nicolas and G. Robin, [250] References 1. S. Ahlgren, Distribution of parity of the partition function in arithmetic pro- gressions, Indag. Math. 10 (1999), 173–181. 2. S. Ahlgren, Non-vanishing of the partition function modulo odd primes ,Math- ematika 46 (1999), 185–192. 3. S. Ahlgren, The partition function modulo composite integers M, Math. Ann. 318 (2000), 795–803. 4. S. Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan’s theta function, Proc. Amer. Math. Soc. 128 (2000), 1333–1338. 5. S. Ahlgren and M. Boylan, Arithmetic properties of the partition function, Invent. Math. 153 (2003), 487–502. 6. S. Ahlgren and M. Boylan, Coefficients of half-integral weight modular forms modulo , Math. Ann. 331 (2005), 219–241. 7. S. Ahlgren and K. Ono, Addition and counting: The arithmetic of partitions, Notices Amer. Math. Soc. 48 (2001), 978–984. 8. S. Ahlgren and S. Treneer, Rank generating functions as weakly holomorphic modular forms,ActaArith.133 (2008), 267–279. 9. L. Alaoglu and P. Erd˝os, On highly composite and similar numbers,Trans. Amer. Math. Soc. 56 (1944), 448–469. 10. A. Alaca, S. Alaca, M.F. Lemire, and K.S. Williams, Nineteen quatenary quadratic forms,ActaArith.130 (2007), 277–310. 11. A. Alaca, S. Alaca, and K.S. Williams, The simplest proof of Jacobi’s six squares theorem, Far East J. Math.
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