George E. Andrews, B..

438 Pages·2012·3.99 MB·English

Ramanujan’s Lost Notebook
George E. Andrews (cid:2) 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
LibraryofCongressControlNumber:2005923547
MathematicsSubjectClassification(2010): 11P82,11P81,11P84,11P83,33Dxx
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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
Thisisthethirdoffivevolumesthattheauthorsplantowriteintheirexami-
nationofalltheclaimsmadebyS.RamanujaninTheLostNotebookandOther
Unpublished Papers,whichwaspublishedbyNarosain1988.Thispublication
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
wrotetoG.H.Hardyfromnursinghomesduring1917–1919.Ourthirdvolume
containstenchaptersandfocusesonsomeofthemostimportantandinfluen-
tial material in The Lost Notebook and Other Unpublished Papers.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 Preliminary Results...................................... 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 Key Formulas on Page 59................................. 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 λ on Pages 179 and 180 .... 75
n
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
x Contents
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) by 5 .................................. 95
5.3 The Congruence p(25n+24)≡0(mod25) .................. 97
5.4 Congruences Modulo 5k .................................. 98
5.5 Congruences Modulo 7 ................................... 99
5.6 Congruences Modulo 7, Continued.........................100
5.7 Congruences Modulo 49 ..................................102
5.8 Congruences Modulo 49, Continued........................104
5.9 The Congruence p(11n+6)≡0(mod11) ...................106
5.10 Congruences Modulo 11, Continued........................107
5.11 Divisibility by 2 or 3.....................................110
5.12 Divisibility of τ(n).......................................113
5.13 Congruences Modulo 13 ..................................119
5.14 Congruences for p(n) Modulo 13 ..........................121
5.15 Congruences to Further Prime Moduli .....................123
5.16 Congruences for p(n) Modulo 17, 19, 23, 29, or 31 ...........125
5.17 Divisibility of τ(n) by 23 .................................127
5.18 The Congruence p(121n−5)≡0(mod121) .................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(mod25), Revisited.........134
5.22 Congruences for p(n) Modulo Higher Powers of 5 ............135
5.23 Congruences for p(n) Modulo Higher Powers of 5, Continued..136
5.24 The Congruence p(7n+5)≡0(mod7) .....................138
5.25 Commentary............................................139
5.1 The Congruence p(5n+4)≡0(mod5) .....................140
5.2 Divisibility of τ(n) by 5 ..................................141
5.4 Congruences Modulo 5k ..................................143
5.5 Congruences Modulo 7 ...................................144
5.6 Congruences Modulo 7, Continued.........................144
5.7 Congruences Modulo 49 ..................................144
5.8 Congruences Modulo 49, Continued........................145
5.9 The Congruence p(11n+6)≡0(mod11) ...................145
5.10 Congruences Modulo 11, Continued........................146
5.11 Divisibility by 2 or 3.....................................147
5.12 Divisibility of τ(n).......................................150
5.13 Congruences Modulo 13 ..................................151
Contents xi
5.14 Congruences for p(n) Modulo 13 ..........................153
5.15 Congruences to Further Prime Moduli .....................153
5.16 Congruences for p(n) Modulo 17, 19, 23, 29, or 31 ...........159
5.17 Divisibility of τ(n) by 23 .................................176
5.18 The Congruence p(121n−5)≡0(mod121) .................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 The Identities for Modulus 5..............................183
6.3 The Identities for Modulus 7..............................186
6.4 Two Beautiful, False, but Correctable Claims of Ramanujan ..193
6.5 Page 182 ...............................................195
6.6 Further Remarks ........................................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
8.2 Definitions and Preliminary Results........................219
8.3 The Forty Identities .....................................222
8.4 The Principal Ideas Behind the Proofs .....................229
8.5 Proofs of the 40 Entries ..................................243
8.6 Other Identities for G(q) and H(q) and Final Remarks .......333
9 Circular Summation .......................................337
9.1 Introduction ............................................337
9.2 Proof of Entry 9.1.1 .....................................339
9.3 Reformulations..........................................342
9.4 Special Cases ...........................................345
10 Highly Composite Numbers ...............................359
Scratch Work..................................................403
Location Guide ................................................407
Provenance ....................................................411
References.....................................................413
1
Introduction
ThisisthethirdvolumedevotedtoRamanujan’slostnotebookandtopartial
manuscripts, fragments, and letters published with the lost notebook [283].
Thecenterpieceofthisvolumeisthe partitionfunctionp(n). Featured inthis
bookarecongruencesforp(n),ranksandcranksofpartitions,theRamanujan
τ-function, the Rogers–Ramanujan functions, and the unpublished portion of
Ramanujan’s paper on highly composite numbers [274].
The first three chapters are devoted to ranks and cranks of partitions. In
1944, F. Dyson [127] defined the rank of a partition to be the largest part
minus the number of parts. If N(m,t,n) denotes the number of partitions of
n with rank congruent to m modulo t, then Dyson conjectured that
p(5n+4)
N(k,5,5n+4)= , 0≤k ≤4, (1.0.1)
5
and
p(7n+5)
N(k,7,7n+5)= , 0≤k ≤6. (1.0.2)
7
Thus,if (1.0.1)and(1.0.2)weretrue,thepartitionscountedbyp(5n+4)and
p(7n+5) would fall into five and seven equinumerous classes, respectively,
thusprovidingcombinatorialexplanationsandproofsforRamanujan’sfamous
congruences p(5n + 4) ≡ 0 (mod 5) and p(7n + 5) ≡ 0 (mod 7). Dyson’s
conjectures were first proved by A.O.L. Atkin and H.P.F. Swinnerton-Dyer
[28] in 1954.
Dyson observed that the corresponding analogue to (1.0.1) and (1.0.2)
does not hold for the third famous Ramanujan congruence p(11n+6) ≡ 0
(mod 11), and so he conjectured the existence of a statistic that he called
thecrank thatwouldcombinatoriallyexplainthiscongruence.Inhisdoctoral
dissertation [144], F.G. Garvan defined a crank for vector partitions, which
becametheforerunnerofthetruecrank,whichwasdiscoveredbyAndrewsand
Garvan [17] during the afternoon of June 6, 1987, at Illinois Street Residence
Hall, a student dormitory at the University of Illinois, following a meeting on
June 1–5 to commemorate the centenary of Ramanujan’s birth.
G.E. Andrews, B.C. Berndt, Ramanujan’s Lost Notebook: Part III, 1
DOI 10.1007/978-1-4614-3810-6 1,
(cid:2)c Springer Science+Business Media New York 2012