RamanuJan Lostnotebook
RamanuJan Lostnotebook
RamanuJan Lostnotebook
BRUCE C. BERNDT
he was encouraged, especially by Sir Francis Spring and S. Narayana Aiyar, Chair-
man and Chief Accountant, respectively. Narayana Aiyar held a Masters Degree in
Mathematics and at that time was probably one of the most capable mathematicians
in India [20]. They persuaded Ramanujan to write English mathematicians about his
mathematical discoveries. One of them, G. H. Hardy, responded encouragingly and
invited Ramanujan to come to Cambridge to develop his mathematical gifts. Ramanu-
jans family were Iyengars, a conservative orthodox branch in the Brahmin tradition,
and especially his mother was adamantly opposed to her sons crossing the seas and
thereby becoming unclean. After overcoming family reluctance, Ramanujan boarded
a passenger ship for England on March 17, 1914.
At about this time, Ramanujan evidently stopped recording his theorems in note-
books. That Ramanujan no longer concentrated on logging entries in his notebooks
is evident from two letters that he wrote to friends in Madras during his first year in
England [31, pp. 112113; 123125]. In a letter of November 13, 1914 to his friend
R. Krishna Rao, Ramanujan confided, I have changed my plan of publishing my re-
sults. I am not going to publish any of the old results in my notebooks till the war is
over. And in a letter of January 7, 1915 to S. M. Subramanian, Ramanujan admitted,
I am doing my work very slowly. My notebook is sleeping in a corner for these four
or five months. I am publishing only my present researches as I have not yet proved
the results in my notebooks rigorously.
On March 24, 1915, near the end of his first winter in Cambridge, Ramanujan wrote
to his friend E. Vinayaka Row [31, pp. 116117] in Madras, I was not well till the
beginning of this term owing to the weather and consequently I couldnt publish any
thing for about 5 months. By the end of his third year in England, Ramanujan was
critically ill, and, for the next two years, he was confined to nursing homes. After World
War I ended, Ramanujan returned home in March, 1919, but his health continued to
deteriorate, and on April 26, 1920 Ramanujan died at the age of 32.
In both England and India, Ramanujan was treated for tuberculosis, but his symp-
toms did not match those of the disease. More recently, an English physician, D. A. B.
Young [32, pp. 6575] carefully examined all extant records and symptoms of Ra-
manujans illness and convincingly concluded that Ramanujan suffered from hepatic
amoebiasis, a parasitic infection of the liver. Amoebiasis is a protozoal infection of
the large intestine that gives rise to dysentery. Relapses occur when the hostparasite
relationship is disturbed, which likely happened when Ramanujan entered a colder
climate. The illness is very difficult to diagnose, but once diagnosed, it can be cured.
Despite being confined to nursing homes for two of his five years in England, Ramanu-
jan made enormously important contributions to mathematics, several in collaboration
with Hardy, which, although they won him immediate and lasting fame, are proba-
bly recognized and appreciated more so today than they were at that time. Most of
Ramanujans discoveries lie in the areas of (primarily) number theory, analysis, and
combinatorics, but they influence many modern branches of both mathematics and
physics.
RAMANUJAN, HIS LOST NOTEBOOK, ITS IMPORTANCE 3
hundred formulas without proofs. Although technically not a notebook, and although
technically not lost, as we shall see in the sequel, it was natural in view of the
fame of Ramanujans (earlier) notebooks [62] to call this manuscript Ramanujans lost
notebook. Almost certainly, this manuscript, or at least most of it, was written during
the last year of Ramanujans life, after his return to India from England.
The manuscript contains no introduction or covering letter. In fact, there are hardly
any words in the manuscript. There are a few marks evidently made by a cataloguer,
and there are also a few remarks in the handwriting of Hardy. Undoubtedly, the most
famous objects examined in the lost notebook are the mock theta functions, about
which more will be written later.
The natural, burning question now is: How did this manuscript of Ramanujan come
into Watsons possession? We think that the manuscripts history can be traced.
that he believes that Ramanujan was unaware of certain third order mock theta func-
tions and transformation formulas for the fifth order mock theta functions. But, in his
lost notebook, Ramanujan did indeed examine these third order mock theta functions,
and found transformation formulas for the fifth order mock theta functions. Watsons
interest in Ramanujans mathematics waned in the late 1930s, and Hardy died in 1947.
In conclusion, sometime between 1934 and 1947 and probably closer to 1947, Hardy
gave Watson the manuscript that we now call the lost notebook.
Watson was Mason Professor of Pure Mathematics at the University of Birmingham
for most of his career, retiring in 1951. He died in 1965 at the age of 79. Rankin,
who succeeded Watson as Mason Professor but who had since become Professor of
Mathematics at the University of Glasgow, was asked to write an obituary of Watson
for the London Mathematical Society. Rankin wrote [65], [32, p. 120]:
For this purpose I visited Mrs Watson on 12 July 1965 and was shown
into a fair-sized room devoid of furniture and almost knee-deep in manu-
scripts covering the floor area. In the space of one day I had time only
to make a somewhat cursory examination, but discovered a number
of interesting items. Apart from Watsons projected and incomplete
revision of Whittaker and Watsons Modern Analysis in five or more
volumes, and his monograph on Three decades of midland railway lo-
comotives, there was a great deal of material relating to Ramanujan,
including copies of Notebooks 1 and 2, his work with B. M. Wilson on
the Notebooks and much other material. . . . In November 19 1965 Dr
J. M. Whittaker who had been asked by the Royal Society to prepare an
obituary notice [72], paid a similar visit and unearthed a second batch
of Ramanujan material. A further batch was given to me in April 1969
by Mrs Watson and her son George.
Since her late husband had been a Fellow and Scholar at Trinity College and had had
an abiding, lifelong affection for Trinity College, Mrs. Watson agreed with Rankins
suggestion that the library at Trinity College would be the most appropriate place to
preserve her husbands papers. Since Ramanujan had also been a Fellow at Trinity
College, Rankins suggestion was even more appropriate.
During the next three years, Rankin sorted through Watsons papers, and dispatched
Watsons and Ramanujans papers to Trinity College in three batches on November
2, 1965; December 26, 1968; and December 30, 1969, with the Ramanujan papers
being in the second shipment. Rankin did not realize the importance of Ramanujans
papers, and so when he wrote Watsons obituary [64] for the Journal of the London
Mathematical Society, he did not mention any of Ramanujans manuscripts. Thus, for
almost eight years, Ramanujans lost notebook and some fragments of papers by
Ramanujan lay in the library at Trinity College, known only to a few of the librarys
cataloguers, Rankin, Mrs. Watson, Whittaker, and perhaps a few others. The 138-page
manuscript waited there until Andrews found it and brought it before the mathematical
public in the spring of 1976. It was not until the centenary of Ramanujans birth on
December 22, 1987, that Narosa Publishing House in New Delhi published in photocopy
form Ramanujans lost notebook and his other unpublished papers [63].
RAMANUJAN, HIS LOST NOTEBOOK, ITS IMPORTANCE 7
scattered sheets and fragments. The three most famous of these unpublished manu-
scripts are those on the partition function and Ramanujans tau function [30], [6], forty
identities for the RogersRamanujan functions [26], [6], and the unpublished remainder
of Ramanujans published paper on highly composite numbers [58], [61, pp. 78128],
[6].
In the passages that follow, we select certain topics and examples to illustrate the
content and importance of Ramanujans discoveries found in his lost notebook. An-
drews and the author are in the process of writing five volumes on the lost notebook
that are analogous to those that the author wrote on the earlier notebooks [14][18].
At the moment of this writing, four of the volumes [4][7] have been published.
Perhaps the most useful property of theta functions is the famous Jacobi triple product
identity [16, p. 35, Entry 19] given by
f (a, b) = (a; ab) (b; ab) (ab; ab) , |ab| < 1, (7.3)
where
(a; q) := lim (a; q)n , |q| < 1.
n
For this exposition, in Ramanujans notation and with the use of (7.3), only one special
case,
X
2
f (q) := f (q, q ) = (1)n q n(3n1)/2 = (q; q) , (7.4)
n=
is relevant for us. If q = e2i , where Im > 0, then q 1/24 f (q) = ( ), the Dedekind
eta function. The last equality in (7.4) renders Eulers pentagonal number theorem.
For a more detailed introduction to q-series, see the authors paper [19].
RAMANUJAN, HIS LOST NOTEBOOK, ITS IMPORTANCE 9
and
(q 5 ; q 5 ) (q 5 ; q 10 )
f0 (q) + 2(q 2 ) = , (8.5)
(q; q 5 ) (q 4 ; q 5 )
where
2 2
X qn X q 5n
f0 (q) := , (q) := 1 + .
n=0
(q; q)n n=0
(q; q 5 )n+1 (q 4 ; q 5 )n
All three functions (q), f0 (q), and (q) are mock theta functions. Both of these
identities have interesting implications in the theory of partitions, which we address in
Section 9.
The discovery of the lost notebook by Andrews and then the publication of the lost
notebook by Narosa in 1988 [63] stimulated an enormous amount of research on mock
theta functions, as researchers found proofs of the many mock theta function identities
found in the lost notebook. We mention only a few of the more important contributions
by: Andrews [2], [3], Andrews and F. Garvan [9], and D. Hickerson [52] on fifth order
mock theta functions; Andrews and Hickerson [10] on sixth order mock theta functions;
Andrews [2] and Hickerson [53] on seventh order mock theta functions; and Y.S. Choi
[37][40] on tenth order mock theta functions.
As we have seen, Ramanujans definition of a mock theta function is somewhat
vague. Can a precise, coherent theory be developed and find its place among the other
great theories of our day? In 1987, at a meeting held at the University of Illinois
commemorating Ramanujan on the centenary of his birth, F. J. Dyson addressed this
question [42, p. 20], [32, p. 269].
The mock theta-functions give us tantalizing hints of a grand synthe-
sis still to be discovered. Somehow it should be possible to build them
into a coherent group-theoretical structure, analogous to the structure
of modular forms which Hecke built around the old theta-functions of
Jacobi. This remains a challenge for the future. My dream is that I
will live to see the day when our young physicists, struggling to bring
the predictions of superstring theory into correspondence with the facts
of nature, will be led to enlarge their analytic machinery to include not
only theta-functions but mock theta-functions. Perhaps we may one day
see a preprint written by a physicist with the title Mock Atkin-Lehner
Symmetry. But before this can happen, the purely mathematical ex-
ploration of the mock-modular forms and their mock-symmetries must
be carried a great deal farther.
Since we invoke the words, modular form, at times in the sequel, we provide here a
brief definition. Let V ( ) = (a + b)/(c + d), where a, b, c, and d are integers such
that ad bc = 1, and where Im > 0. Then f ( ) is a modular form of weight k if
f (V ( )) = (a, b, c, d)(c + d)k f ( ), (8.6)
where k is real (usually an integer or half of an integer) and |(a, b, c, d)| = 1.
In recent years, the work of S. Zwegers [73] and K. Bringmann and K. Ono [35],
[36] has made progress in the direction envisioned by Dyson. First it was observed, to
take one example, that the infinite product on the right-hand side of (8.5) essentially
RAMANUJAN, HIS LOST NOTEBOOK, ITS IMPORTANCE 11
coincided with the Fourier expansion of a certain weakly holomorphic modular form,
where the term weakly holomorphic indicates that the modular form is analytic in
the upper half-plane, but may have poles at what are called cusps on the real axis.
In his doctoral dissertation [73], Zwegers related mock theta functions to real analytic
vector-valued modular forms by adding to Ramanujans mock theta functions certain
non-holomorphic functions, which are called period integrals. Earlier work of Andrews
[2], who used Bailey pairs to express Ramanujans Eulerian series in terms of Hecke-
type series, was also essential for Zwegers, since he applied his ideas to the Hecke-type
series rather than Ramanujans original series. Zwegers real analytic modular forms
are examples of harmonic Maass forms. Briefly, a Maass form satisfies the functional
equation (8.6) with k = 0 and is an eigenfunction of the hyperbolic Laplacian
2
2
2
:= y + , (8.7)
x2 y 2
where = x + iy. A harmonic Maass form M ( ) again satisfies a functional equation
of the type (8.6) when k is an integer and a slightly different functional equation if k
is half of an integer, but the operator (8.7) is replaced by
2
2
2
k := y + + iky +i .
x2 y 2 x y
It is now required that k M = 0.
Returning to (8.2), Bringmann and Ono [36] examined the more general function
2
X qn
R(, q) := (8.8)
n=0
(q; q)n ( 1 q; q)n
and proved (under certain hypotheses), when is a root of unity, that R(, q) is the
holomorphic part of a weight 21 harmonic Maass form. In general, Bringmann and Ono
showed that each of Ramanujans mock theta functions is the holomorphic part of a
harmonic Maass form.
Recall that in their famous paper [50], Hardy and Ramanujan developed an as-
ymptotic series for the partition function p(n), defined to be the number of ways the
positive integer n can be expressed as a sum of positive integers. For example, since
4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1, p(4) = 5. Some years later, improving
on their work, H. Rademacher [57] found a convergent series representation for p(n).
If we write the mock theta function f (q) from (8.2) as f (q) = n
P
n=0 (n)q , Andrews
[1] analogously found an asymptotic series for (n). Bringmann and Ono [35] were
able to replace the asymptotic formula by an exact formula confirming a conjecture of
Andrews.
We have sketched only a few highlights among the extensive recent developments in
the theory of mock theta functions. Readers are encouraged to read Onos compre-
hensive description [56] of these developments. In Onos paper, readers will also find
discussions and references to the permeation of mock theta functions in physics, thus
providing evidence for Dysons prophetic vision.
12 BRUCE C. BERNDT
9. Partitions
Recall from above the definition of the partition function p(n). Inspecting a table of
p(n), 1 n 200, calculated by P. A. MacMahon, Ramanujan was led to conjecture
the congruences
p(5n + 4) 0 (mod 5),
p(7n + 5) 0 (mod 7), (9.1)
p(11n + 6) 0 (mod 11),
which he eventually proved [60], [63], [30]. In 1944, Dyson [41] sought to combinatori-
ally explain (9.1) and in doing so defined the rank of a partition to be the largest part
minus the number of parts. For example, the rank of 3+1 is 1. Dyson observed that the
congruence classes for the rank modulo 5 and 7 divided the partitions of p(5n + 4) and
p(7n+5), respectively, into equinumerous classes. These conjectures were subsequently
proved by A. O. L. Atkin and H. P. F. Swinnerton-Dyer [11]. However, for the third
congruence in (9.1), the corresponding criterion failed, and so Dyson conjectured the
existence of a statistic, which he called the crank to combinatorially explain the con-
gruence p(11n + 6) 0 (mod 11). The crank of a partition was found by Andrews and
Garvan [8] and is defined to be the largest part if the partition contains no ones, and
otherwise to be the number of parts larger than the number of ones minus the num-
ber of ones. The crank divides the partitions into equinumerous congruence classes
modulo 5, 7, and 11 for the three congruences, respectively, in (9.1).
At roughly the same time that Andrews and Garvan found the crank, it was observed
that in his lost notebook, Ramanujan had found the generating functions for both the
rank and the crank. First, if N (m, n) denotes the number of partitions of n with rank
m, then
2
X X
m n
X qn
N (m, n)z q = . (9.2)
n=0 m= n=0
(zq; q)n (z 1 q; q)n
The definition (9.2) should be compared with that in (8.8). Second, if M (m, n) denotes
the number of partitions of n with crank m, then, except for a few small values of m
and n,
X X (q; q)
M (m, n)z m q n = . (9.3)
n=0 m=
(zq; q) (z 1 q; q)
We do not know if Ramanujan knew the combinatorial implications of the rank and
crank. However, in view of the several results on the generating functions for the rank
and crank as well as calculations for cranks found in his lost notebook, it is clear that
he had realized the importance of these two functions [46], [23]. There is also evidence
that his very last mathematical thoughts were on cranks before he died on April 26,
1920 [24].
Many identities in the lost notebook have partition theoretic implications. As
promised earlier, we now examine the partition-theoretic interpretations of (8.4) and
(8.5).
We state (8.4) in an equivalent form: The number of partitions of a positive integer
N where the smallest part does not repeat and the largest part is at most twice the
RAMANUJAN, HIS LOST NOTEBOOK, ITS IMPORTANCE 13
smallest part equals the number of partitions of N where the largest part is odd and
the smallest part is larger than half the largest part. As an example, take N = 7. Then
the relevant partitions are, respectively, 7 = 4 + 3 = 2 + 2 + 2 + 1 and 7 = 3 + 2 + 2 =
1 + 1 + 1 + 1 + 1 + 1 + 1. A short proof can be constructed with the use of Ferrers
diagrams, and we leave this proof for the reader.
To examine (8.5), we first define 0 (n) to be the number of partitions of n with
unique smallest part and all other parts the double of the smallest part. For example,
0 (5) = 3, with the relevant partitions being 5, 3+2, and 2+2+1. Second, let N (a, b, n)
denote the number of partitions of n with rank congruent to a modulo b. Then (8.5)
is equivalent to The First Mock Theta Conjecture,
For example, if n = 5, then N (1, 5, 25) = 393, N (0, 5, 25) = 390, and, as observed
above, 0 (5) = 3. Although (8.5) has been proved by Hickerson, and now also by
A. Folsom [43] and by Hickerson and E. Mortenson [54], a combinatorial proof of (9.4)
has never been given.
where |t|, |b| < 1. There are many identities in the lost notebook, whose proofs naturally
use Heines transformation [5, Chapter 1]. One consequence of (10.1) is, for |aq| < 1
[63, p. 38],
X (aq)n X (1)n an q n(n+1)/2
2; q2)
= . (10.2)
n=0
(aq n n=0
(aq; q) n
To see how (10.1) can be used to prove (10.2), consult [5, p. 25, Entry 1.6.4]. For
partition-theoretic proofs, see [33] and [27].
Our second identity is given by [63, p. 35], [5, p. 35, Entry 1.7.9]
X (1)n q n(n+1)/2
2n+1
= f (q 3 , q 5 ), (10.3)
n=0
(q; q)n (1 q )
where f (a, b) is defined by (7.2). We see partitions at work on the left-hand side of
(10.3), but on the right-hand side, we observe that these partitions have cancelled each
other out, except on a thin set of quadratic exponents. For a combinatorial proof of
(10.3), see [27].
14 BRUCE C. BERNDT
as x , where the prime 0 on the summation sign on the left side indicates that if
x is an integer, only 21 r2 (x) is counted. We now explain why this problem is called the
circle problem. Each representation of n as a sum of two squares can be associated with
a lattice point in the plane. For example, 5 = (2)2 + 12 can be associated with the
lattice point (2, 1). Then each lattice point can be associated with a unit square, say
that unit square for which the lattice point is in the southwest corner. Thus, the sum
in (13.1) is equal to the number of lattice points in the circle of radius x centered at
the origin, or to the sum of the areas of the aforementioned squares, and the area of
this circle, namely x, is a reasonable approximation to the sum of the areas of these
rectangles. Gauss showed quite easily that the error made in this approximation is
P (x) = O( x).
In 1915, Hardy [49] proved that P (x) 6= O(x1/4 ), as x tends to . In other words,
1/4
there is a sequence of points {xn } tending to on which P (xn ) 6= O(xn ). (He
actually proved a slightly stronger result.) In connection with his work on the circle
problem, Hardy [49] proved that
X 0 X x 1/2
r2 (n) = x + r2 (n) J1 (2 nx), (13.2)
0nx n=1
n
where J1 (x) is the ordinary Bessel function of order 1. After Gauss, almost all efforts
toward obtaining an upper bound for P (x) have ultimately rested upon (13.2), and
methods of estimating the approximating
trigonometric series that is obtained from
the asymptotic formula for J1 (2 nx) as n . In 1906, W. Sierpinski [68] proved
that P (x) = O(x1/3 ) as x tends to , and there have been many improvements in a
century of work since then, with the best current result being P (x) = O(x131/416+ ),
for every > 0, due to M. N. Huxley [55] ( 131 416
= 0.3149 . . . ). It is conjectured that
1/4+
P (x) = O(x ), for every > 0. In a footnote, Hardy remarks, The form of this
equation was suggested to me by Mr. S. Ramanujan, to whom I had communicated
the analogous formula for d(1) + d(2) + + d(n), where d(n) is the number of divisors
of n. In this same paper, Hardy relates a beautiful identity of Ramanujan connected
with r2 (n), namely, for a, b > 0, [49, p. 283],
Entry 13.1 was first proved by the author, S. Kim, and A. Zaharescu [29]. In [28],
the same three authors proved (13.4) with the product mn of the summation indices
tending to . Entry 13.1 was first established with the order of summation reversed
from that prescribed by Ramanujan in (13.4) [34].
The Bessel functions in (13.4) bear a striking resemblance to those in (13.2), and
so it is natural to ask if there is a connection between the two formulas. Berndt,
Kim, and Zaharescu [28] proved the following corollary, as a consequence of their
reinterpreted meaning of the double sum. It had been previously established (although
not rigorously) by Berndt and Zaharescu in [34] as a corollary of their theorem on
twisted divisor sums arising from Entry 13.1.
Corollary 13.2. For any x > 0,
q q
X
1
J1 4 m(n + 4 )x 3
X 0 X
J1 4 m(n + 4 )x
r2 (n) = x + 2 x q q .
1 3
0nx n=0 m=1
m(n + 4
) m(n + 4
)
(13.5)
Can Corollary 13.2 be employed in place of (13.2) to effect an improvement in the
error term for the circle problem? The advantage of (13.5) is that r2 (n) does not
appear on the right-hand side; the disadvantage is that one needs to estimate a double
sum, instead of a single sum in (13.2).
The second identity on page 335 pertains to the famous Dirichlet divisor problem.
Let d(n) denote the number of positive divisors of the integer n. Define the error
term (x), for x > 0, by
X0 1
d(n) = x (log x + (2 1)) + + (x), (13.6)
nx
4
where denotes Eulers constant, and where the prime 0 on the summation sign on
the left side indicates that if x is an integer, then only 12 d(x) is counted. The famous
Dirichlet divisor problem asks for the correct order of magnitude of (x) as x .
Counting lattice points under a hyperbola, Dirichlet easily showed (13.6) with (x) =
18 BRUCE C. BERNDT
O( x). Estimates that have been established for the error term (x) are similar to
those that have been proved for P (x). G. F. Vorono [69] established a representation
for (x) in terms of Bessel functions with his famous formula
X0 1 X x 1/2
d(n) = x (log x + (2 1)) + + d(n) I1 (4 nx), (13.7)
nx
4 n=1 n
where x > 0 and
2
I (z) := Y (z)
K (z). (13.8)
Here Y1 (x) denotes the Bessel function of the second kind of order 1, and K1 (x) denotes
the modified Bessel function of order 1. Since its appearance in 1904, (13.7) has been
the starting point for nearly all attempts at finding an upper bound for (x).
We close this section by stating the second identity on page 335 of [63]. Note that
the Bessel functions that appear below are the same as those in (13.7). Proofs of Entry
13.3 with the order of summation reversed from that given by Ramanujan in (13.9)
and with the product of the summation indices m and n approaching can be found
in the authors paper with Kim and Zaharescu [28]. See also [7, Chapter 2].
Entry 13.3 (p. 335). Let F (x) be defined by (13.3), and let I1 (x) be defined by (13.8).
Then, for x > 0 and 0 < < 1,
x
X 1
F cos(2n) = x log(2 sin())
n=1
n 4
p p
1 X X I
1 4 m(n + )x I1 4 m(n + 1 )x
+ x p + p . (13.9)
2 m=1 n=0
m(n + ) m(n + 1 )
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RAMANUJAN, HIS LOST NOTEBOOK, ITS IMPORTANCE 19
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