topics: |  mathematics | biography | ramanujan |

In the shadow of two cultures

They met under the severe asymmetry of colonialism.  The British were the
colonial rulers of India, justifying their rule in terms of their manifest
destiny to improve the globe. 

Under these circumstances, an unknown Indian writes a letter to two
mathematicians at Cambridge - E.H. Neville and G.H. Hardy.  A long list of 
results are attached.  They see
a spark somewhere, that they cannot explain.  Hardy noted later that
it could not have been a charlatan, since "great mathematicians are
commoner than humbugs of such incredible skill."


     the encounter of ramanujan and hardy is one of the enduring
     romantic stories in mathematics.

They bring Ramanujan to Cambridge, and the clash of cultures continues.
He is a strict vegetarian, a devotee of goddess Namakkal - the
deity Namagiri Lakshmi at the district town of Namakkal, not far from
Trichy.

At Cambridge, he cooks his own food. Hardy tells us that "he never cooked
without first changing into pyjamas."  We presume he would also have had a
quick bath and a small cleansing puja (achman).

The mystery of a colonial encounter

Given this lifestyle, Hardy is surprised one day, when ramanujan tells him
"that all religions seemed to him more or less equally true."

Hardy takes pains to tell us that this was no sophistry on the part of
Ramanujan, for he was very direct in all such matters.  Therefore religion,
concludes Hardy,
	except in a strictly material sense, played no important part in his
	life.

Steeped in his own religious culture, Hardy is unable to understand a
tradition that accepts multiplicity of religious thought.  You can have
your own gods, and I can have mine.

Hardy appears to interpret this statement as that of holding no belief:
	if a strict Brahmin like Ramanujan told me, as he
	certainly did, that he had no definite beliefs, then it is
	100 to 1 that he meant what he said.

So we find that "all religions are equally true" is taken to be equivalent
form for the statement "he had no definite beliefs".

Based on this, Hardy is convinced that Ramanujan was merely an "observing
Hindu", not a true believer:
     	...  his religion was a matter of observance and not of intellectual
     	conviction... He was not a reasoned infidel, but an
	"agnostic " in its strict sense, who saw no particular good, and no
	particular harm, in Hinduism or in any other religion.

In this manner Hardy seems to be reconstructing Ramanujan into the
straitjacket of his own cultural milieu.  On the basis of this
understanding he challenges the views of Seshu Aiyar and Ramachaundra Rao,
who had known Ramanujan in Madras:

    Ramanujan had definite religious views. He had a special veneration
    for the Namakkal goddess He believed in the existence of a Supreme
    Being and in the attainment of Godhead by men...  He had settled
    convictions about the problem of life and after...;

Hardy comments on this difference:
	Which of us is right? For my part I have no doubt at all; I am quite
	certain that I am....

This type of cultural misunderstanding apart, the mathematics of the
interaction speaks in its own language, where too there are many
misunderstandings, owing to the very different background that Ramanujan
had.  But here, Hardy is on much more solid ground in terms of interpreting
the provenance of his find.

Ramanujan's mathematics: The Partition function

Most of us, even those who have had a reasonably good mathematical
education, are quite lost when it comes to judge Ramanujan's quality. 

Here is a simple example, that most people can grasp. 

The partition function p(n) is the number of ways the given number _n_ can
be expressed as the sum of other integers.  Thus, 4 can be
written as: 
	4 = 4, 
	  = 3+1, 
	  = 2+2, 
	  = 2+1+1, 
	  = 1+1+1+1;  
so  P(4)=5.  

The partition function grows very rapidly. While p(10)=42, 
p(100) = 190,569,292, and .  Earlier, Euler and others had proposed complex recurrence
relations (where you do long infinite sums to get the value of P(n).  But
Ramanujan came up with the approximation:

		

With some help from Hardy, they were able to show (1918) that this result
follows from some theorems in modular functions.  But, to check the theorem a
fellow Cambridge mathematician tallied by hand the partitions for 200.  It
took one month [p(200)=397,999,029,388], and the formula was amazingly
accurate!  

Many have commented that Ramanujan's numerical accuracy was unprecedented -
matched perhaps only by Euler or Jacobi.

Looking at the formula above, we find _e_ raised to pi multiplied by sqrt(n).
For uninitiated folks like me, the formula is amazingly elegant - somewhat
reminescent of the equation e^{i\pi} = -1. 

One can never cease to wonder how such insights could have come to
Ramanujan.  Reading his biography, together with other calculating
prodigies and other gifted individuals, sometimes one thinks that for some
types of minds (or maybe all?), too much education may actually be a
handicap...

Excerpts

[Ramanujan was] the most romantic figure in the recent history of
mathematics; a man whose career seems full of paradoxes and contradictions,
who defies almost all the canons by which we are accustomed to judge one
another, and about whom all of us will probably agree in one judgment only,
that he was in some sense a very great mathematician.

They must be true because, if they were not true, no one would have had the imagination to invent them.
- Hardy, on results sent by Ramanujan.

The real difficulty for me is that Ramanujan was, in a way, my discovery. I did not invent him — like other great men, he invented himself — but I was the first really competent person who had the chance to see some of his work, and I can still remember with satisfaction that I could recognise at once what a treasure I had found. And I suppose that I still know more of Ramanujan than any one else, and am still the first authority on this particular subject. There are other people in England, Professor Watson in particular, and Professor Mordell, who know parts of his work very much better than I do, but neither Watson nor Mordell knew Ramanujan himself as I did. I saw him and talked with him almost every day for several years, and above all I actually collaborated with him. I owe more to him than to anyone else in the world with one exception, and my association with him is the one romantic incident in my life. The difficulty for me then is not that I do not know enough about him, but that I know and feel too much and that I simply cannot be impartial.

I rely, for the facts of Ramanujan's life, on Seshu Aiyar and Ramachaundra
Rao, whose memoir of Ramanujan is printed, along with my own,
in his Collected Papers. He was born in 1887 in a Brahmin family at Erode
near Kumbakonam, a fair-sized town in the Tanjore district of the
Presidency of Madras. His father was a clerk in a cloth-merchant's office in
Kumbakonam, and all his relatives, though of high caste, were very poor.

He was sent at seven to the High School of Kumbakonam, and remained there
nine years. His exceptional abilities had begun to show themselves before he
was ten, and by the time that he was twelve or thirteen he was recognised as
a quite abnormal boy.

His biographers tell some curious stories of his early years.  They say, for
example, that soon after he had begun the study of trigonometry, he
discovered for himself "Euler's theorems for the sine and cosine" (by which I
understand the relations between the circular and exponential functions), and
was very disappointed when he found later, apparently from the second volume
of Loney's Trigonometry, that they were known already. Until he was sixteen
he had never seen a mathematical book of any higher class. Whittaker's Modern
analysis had not yet spread so far, and Bromwich's Infinite series did not
exist. There can be no doubt that either of these books would have made a
tremendous difference to him if they could have come his way. It was a book
of a very different kind, Carr's Synopsis, which first aroused Ramanujan's
full powers.

Ramanujan's sources: Carr's synopsis

Carr's book (A synopsis of elementary results in pure and applied
mathematics, by George Shoobridge Carr, formerly Scholar of Gonville and
Caius College, Cambridge, published in two volumes in 1880 and 1886) is
almost unprocurable now. There is a copy in the Cambridge University
Library, and there happened to be one in the library of the Government
College of Kumbakonam, which was borrowed for Ramanujan by a friend. The
book is not in any sense a great one, but Ramanujan has made it famous, and
there is no doubt that it influenced him profoundly and that his
acquaintance with it marked the real starting-point of his career.

Carr's work, if not a book of any high, distinction, is no mere third-rate
textbook, but a book written with some real scholarship and enthusiasm and
with a style and individuality of its own.  Carr himself was a private
coach in London, who came to Cambridge as an undergraduate when he was
nearly forty, and was 12th Senior Optime in the Mathematical Tripos of
1880 (the same year in which he published the first volume of his
book). He is now completely forgotten, even in his own college, except in
so far as Ramanujan has kept his name alive; but he must have been in some
ways rather a remarkable man.

... great mathematicians are commoner than thieves or humbugs of such incredible skill.

I suppose that the book is substantially a summary of Carr's coaching notes. If you were a pupil of Carr, you worked through the appropriate sections of the Synopsis. It covers roughly the subjects of Schedule A of the present Tripos (as these subjects were understood in Cambridge in 1880), and is effectively the " synopsis " it professes to be. It contains the enunciations of 6165 theorems, systematically and quite scientifically arranged, with proofs which are often little more than cross-references and are decidedly the least interesting part of the book. All this is exaggerated in Ramanujan's famous notebooks (which contain practically no proofs at all), and any student of the notebooks can see that Ramanujan's ideal of presentation had been copied from Carr's.

... some sections are developed disproportionally, and particularly the
formal side of the integral calculus. This seems to have been Carr's pet
subject, and the treatment of it is very full and in its way definitely
good. There is no theory of functions; and I very much doubt whether
Ramanujan, to the end of his life, ever understood at all clearly what an
analytic function is.  3

What is more surprising, in view of Carr's own tastes and Ramanujan's
later work, is that there is nothing about elliptic functions. However
Ramanujan may have acquired his very peculiar knowledge of this theory, it
was not from Carr.

On the whole, considered as an inspiration for a boy of such abnormal
gifts, Carr was not too bad, and Ramanujan responded amazingly.

Seshu Aiyar and Ramachaundra Rao, his Indian biographers:

   Through the new world thus opened to him Ramanujan went ranging with
   delight. It was this book which awakened his genius. He set himself to
   establish the formulae given therein. As he was without the aid of
   other books, each solution was a piece of research so far as he was
   concerned... Ramanujan used to say that the goddess of Namakkal
   inspired him with the formulae in dreams. It is a remarkable fact that
   frequently, on rising from bed, he would note down results and rapidly
   verify them, though he was not always able to supply a rigorous
   proof...

[Hardy is apologetic for having to quote the goddess of Namakkal, but he
does it because] we must try to understand what we can of his psychology
and of the atmosphere surrounding him in his early years.

Ramanujan's religion

I am sure that Ramanujan was no mystic and that religion, except in a
strictly material sense, played no important part in his life. He was an
orthodox high-caste Hindu, and always adhered (indeed with a severity
most unusual in Indians resident in England) to all the observances of his
caste. He had promised his parents to do so, and he kept his promises to
the letter. He was a vegetarian in the strictest sense — this proved a
terrible difficulty later when he fell ill — and - all the time he was in
Cambridge he cooked all his food himself, and never cooked it without first
changing into pyjamas.

Now the two memoirs of Ramanujan printed in the Papers (and both
written by men who, in their different ways, knew him very well) contradict
one another flatly about his religion. Seshu Aiyar and Ramachaundra
Rao say

    Ramanujan had definite religious views. He had a special veneration
    for the Namakkal goddess He believed in the existence of a Supreme
    Being and in the attainment of Godhead by men...  He had settled
    convictions about the problem of life and after...;

while I say

    .. .his religion was a matter of observance and not of intellectual
    conviction, and I remember well his telling me (much to my surprise)
    that all religions seemed to him more or less equally true

Which of us is right? For my part I have no doubt at all; I am quite
certain that I am. ... if a strict Brahmin like Ramanujan told me, as he
certainly did, that he had no definite beliefs, then it is 100 to 1 that
he meant what he said.

[Here Hardy assumes that "all religions seemed to him more or less equally
true" implies a repudiation of religion as a whole.  ]

This was no sufficient reason why Ramanujan should outrage the feelings of
his parents or his Indian friends. He was not a reasoned infidel, but an
"agnostic " in its strict sense, who saw no particular good, and no
particular harm, in Hinduism or in any other religion. Hinduism is, far
more, for example, than Christianity, a religion of observance, in which
belief counts for extremely little in any case, and, if Ramanujan's
friends assumed that he accepted the conventional doctrines of such a
religion, and he did not disillusion them, he was practising a quite
harmless, and probably necessary, economy of truth.

I want to make it quite clear to you that Ramanujan, when he was living in
Cambridge in good health and comfortable surroundings, was, in spite of
his oddities, as reasonable, as sane, and in his way as shrewd a person as
anyone here. [does not want us to associate some kind of mysticism
with Ramanujan...]

the picture which I want to present to you is that of a man who had his peculiarities like other distinguished men, but a man in whose society one could take pleasure, with whom one could drink tea and discuss politics or mathematics; the picture in short, not of a wonder from the East, or an inspired idiot, or a psychological freak, but of a rational human being who happened to be a great mathematician.

Ramanujan after his Matriculation (1903-1913)

Until he was about seventeen, all went well with Ramanujan.

In December 1903 he passed the Matriculation Examination of the University of Madras, and in the January of the succeeding year he joined the Junior First in Arts class of the Government College, Kumbakonam, and won the Subrahmanyam scholarship, which is generally awarded for proficiency in English and Mathematics...,

but after this there came a series of tragic checks.

    By this time, he was so absorbed in the study of Mathematics that in
    all lecture hours — whether devoted to English, History, or Physiology
    — he used to engage himself in some mathematical investigation,
    unmindful of what was happening in the class. This excessive devotion
    to mathematics and his consequent neglect of the other subjects
    resulted in his failure to secure promotion to the senior class and in
    the consequent discontinuance of the scholarship.

    Partly owing to disappointment and partly owing to the influence of a
    friend, he ran away northward into the Telugu country, but returned to
    Kumbakonam after some wandering and rejoined the college. As owing to
    his absence he failed to make sufficient attendances to obtain his
    term certificate in 1905, he entered Pachaiyappa's College, Madras, in
    1906, but falling ill returned to Kumbakonam. He appeared as a private
    student for the F.A. examination of December 1907 and failed....

Ramanujan does not seem to have had any definite occupation, except
mathematics, until 1912. In 1909 he married, and it became necessary for
him to have some regular employment, but he had great difficulty in finding
any because of his unfortunate college career. About 1910 he began to find
more influential Indian friends, Ramaswami Aiyar and his two biographers,
but all their efforts to find a tolerable position for him failed, and in
1912 he became a clerk in the office of the Port Trust of Madras, at a
salary of about £30 a year. He was then nearly twenty-five. The years
between eighteen and twenty-five are the critical years in a
mathematician's career, and the damage had been done. Ramanujan's genius
never had again its chance of full development.

He had an impossible handicap -- a poor and solitary Hindu pitting his brains against the accumulated wisdom of Europe.

There is not much to say about the rest of Ramanujan's life. His first substantial paper had been published in 1911, and in 1912 his exceptional powers began to be understood. It is significant that, though Indians could befriend him, it was only the English who could get anything effective done. Sir Francis Spring and Sir Gilbert Walker obtained a special scholarship for him, £60 a year, sufficient for a married Indian to live in tolerable comfort.

1913: Writes to Hardy. Death 1920

At the beginning of 1913 he wrote to me, and Professor Neville
and I, after many difficulties, got him to England in 1914.  Here he had
three years of uninterrupted activity, the results of which you can read in
the Papers.  He fell ill in the summer of 1917, and never really recovered,
though he continued to work, rather spasmodically, but with no real sign
of degeneration, until his death in 1920.  He became a Fellow of the Royal
Society early in 1918, and a Fellow of Trinity College, Cambridge, later in
the same year (and was the first Indian elected to either society).  His last
mathematical letter on "Mock-Theta functions", the subject of Professor
Watson's presidential address to the London Mathematical Society last
year, was written about two months before he died.

The tragedy of Ramanujan was not that he died young, [Abel died at 26] but
that, during his five unfortunate years, his genius was misdirected,
sidetracked, and to a certain extent distorted.

[16 years ago I had written]:

    Opinions may differ about the importance of Ramanujan's work, the kind
    of standard by which it should be judged, and the influence which it
    is likely to have on the mathematics of the future. It has not the
    simplicity and the inevitableness of the very greatest work; it would
    be greater if it were less strange. One gift it shows which no one can
    deny, profound and invincible originality. He would probably have been
    a greater mathematician if he could have been caught and tamed a
    little in his youth; he would have discovered more that was new, and
    that, no doubt, of greater importance. On the other hand he would have
    been less of a Ramanujan, and more of a European professor, and the
    loss might have been greater than the gain...

I stand by [all that] except for the last sentence, which is quite
ridiculous sentimentalism.

There was no gain at all when the College at Kumbakonam rejected the one
great man they had ever possessed, and the loss was irreparable; it is the
worst instance that I know of the damage that can be done by an
inefficient and inelastic educational system.
So little was wanted, £60 a year for five years, occasional contact with
almost anyone who had real knowledge and a little imagination, for the
world to have gained another of its greatest mathematicians.

15 theoremes from Ramanujan's letters to Hardy

Ramanujan's letters to me, which are reprinted in full in the Papers,
contain the bare statements of about 120 theorems, mostly formal identities
extracted from his notebooks. I quote fifteen which are fairly
representative. They include two theorems, (1.14) and (1.15), which are as
interesting as any but of which one is false and the other, as stated,
misleading. The rest have all been verified since by somebody; in
particular Rogers and Watson found the proofs of the extremely difficult
theorems (1.10)-(1.12).

[list of 15 statements]

I should like you to begin by trying to reconstruct the immediate reactions
of an ordinary professional mathematician who receives a letter like this
from an unknown Hindu clerk.

The first question was whether I could recognise anything. I had proved
things rather like (1.7) myself, and seemed vaguely familiar with (1.8).
Actually (1.8) is classical; it is a formula of Laplace first proved
properly by Jacobi; and (1.9) occurs in a paper published by Rogers in
1907.



I thought that, as an expert in definite integrals, I could probably
prove (1.5) and (1.6), and did so, though with a good deal more trouble
than I had expected. On the whole the integral formulae seemed the least
impressive.



The series formulae (1.1)-(1.4) I found much more intriguing,
and it soon became obvious that Ramanujan must possess much more general
theorems and was keeping a great deal up his sleeve. The second is a
formula of Bauer well known in the theory of Legendre series, but the
others are much harder than they look. The theorems required in proving
them can all be found now in Bailey's Cambridge Tract on hypergeometric
functions.

Continued fractions (Elliptic functions)

The formulae (1.10)—(1.13) are on a different level and obviously both
difficult and deep. An expert in elliptic functions can see at once that
(1.1З) is derived somehow from the theory of "complex multiplication", but
(1.10)-(1.12) defeated me completely; I had never seen anything in the
least like them before.



A single look at them is enough to show that they could only be written
down by a mathematician of the highest class. They must be true because, if
they were not true, no one would have had the imagination to invent them.

Finally (you must remember that I knew nothing whatever about Ramanujan,
and had to think of every possibility), the writer must be completely
honest, because great mathematicians are commoner than thieves or humbugs
of such incredible skill.

The last two formulae stand apart because they are not right and show
Ramanujan's limitations, but that does not prevent them from being
additional evidence of his extraordinary powers. The function in (1.14) is
a genuine approximation to the coefficient, though not at all so close as
Ramanujan imagined, and Ramanujan's false statement was one of the
most fruitful he ever made, since it ended by leading us to all our joint
work on partitions. Finally (1.15), though literally "true", is definitely
misleading (and Ramanujan was under a real misapprehension). The integral
has no advantage, as an approximation, over the simpler function
found in 1908 by Landau.

It was inevitable that a very large part of Ramanujan's work should prove
on examination to have been anticipated. He had been carrying an
impossible handicap, a poor and solitary Hindu pitting his brains against
the accumulated wisdom of Europe. He had had no real teaching at all;
there was no one in India from whom he had anything to learn. He can have
seen at the outside three or four books of good quality, all of them
English.  [==> no French or German]

I should estimate that about two-thirds of Ramanujan's best Indian work was
rediscovery, and comparatively little of it was published in his lifetime,
though Watson, who has worked systematically through his notebooks, has
since disinterred a good deal more.

The great bulk of Ramanujan's published work was done in England.
His mind had hardened to some extent, and he never became at all an
"orthodox" mathematician, but he could still learn to do new things, and
do them extremely well. It was impossible to teach him systematically, but
he gradually absorbed new points of view. In particular he learnt what was
meant by proof, and his later papers, while in some ways as odd and
individual as ever, read like the works of a well-informed mathematician.
His methods and his weapons, however, remained essentially the same. One
would have thought that such a formalist as Ramanujan would have
revelled in Cauchy's Theorem, but he practically never used it, and the
most astonishing testimony to his formal genius is that he never seemed
to feel the want of it in the least.

there is a good deal which we should like to know now and which I could
have discovered quite easily.  I saw Ramanujan almost every day, and could
have cleared up most of the obscurity by a little cross-examination. ... I
hardly asked him a single question of this kind; I never even asked him
whether (as I think he must have done) he had seen Cayley's or Greenhill's
Elliptic functions.

I am sorry about this now, [but] after all I too was a mathematician, and
a mathematician meeting Ramanujan had more interesting things to think
about than historical research.  It seemed ridiculous to worry him about
how he had found this or that known theorem, when he was showing me half a
dozen new ones almost every day.

As Littlewood says, "the clear-cut idea of what is meant
by a proof, nowadays so familiar as to be taken for granted, he perhaps did
not possess at all; if a significant piece of reasoning occurred somewhere,
and the total mixture of evidence and intuition gave him certainty, he
looked no further".

Proof and its importance in mathematics

All physicists, and a good many quite respectable mathematicians, are
contemptuous about proof. I have heard Professor Eddington, for example,
maintain that proof, as pure mathematicians understand it, is really
quite uninteresting and unimportant, and that no one who is really certain
that he has found something good should waste his time looking for a
proof. It is true that Eddington is inconsistent, and has sometimes even
descended to proof himself. It is not enough for hrm to have direct
knowledge that there are exactly
			136.2256
protons in the universe; he cannot resist the temptation of proving it;
and I cannot help thinking that the proof, whatever it may be worth, gives
him a certain amount of intellectual satisfaction. His apology would no
doubt be that "proof" means something quite different for him from what it
means for a pure mathematician, and in any case we need not take him too
literally. p.15

There are a few points about proof where nearly all mathematicians are
agreed. In the first place, even if we do not understand exactly what
proof is, we can, in ordinary analysis at any rate, recognise a proof when
we see one. Secondly, there are two different motives in any presentation
of a proof. The first motive is simply to secure conviction. The second is
to exhibit the conclusion as the climax of a conventional pattern of
propositions, a sequence of propositions whose truth is admitted and which
are arranged in accordance with rules. These are the two ideals, and
experience shows that, except in the simplest mathematics, we can hardly
ever satisfy the first ideal without also satisfying the second.  We may
be able to recognise directly that 5, or even 17, is prime, but nobody can
convince himself that 2127 — 1 is prime except by studying a proof.

The necessity of proof

A mathematician usually discovers a theorem by an effort of intuition;
the conclusion strikes him as plausible, and he sets to work to manufacture
a proof. Sometimes this is a matter of routine, and any well-trained
professional could supply what is wanted, but more often imagination is a very
unreliable guide. In particular this is so in the analytic theory of numbers,
where even Ramanujan's imagination led him very seriously astray.

There is a striking example, which I have very often quoted, of a false
conjecture which seems to have been endorsed even by Gauss and which
took about 100 years to refute. The central problem of the analytic theory
of numbers is that of the distribution of the primes. The number π(χ) of
primes less than a large number x is approximately x/lnx.

This is the "Prime Number Theorem", which had been conjectured for a
very long time, but was never established properly until Hadamard and
de la Vallee-Poussin proved it in 1896.

The approximation is a lower bound, and there exists another function, the
logarithmic integral (integration from zero to infty of x/lnx),
 which is a better estimate, from above.


   plot of pi(x) as a ratio with x/lnx and the offset
   logarithmic integral, (Note that the integral from 2
   to infty, whereas li(x) is the integral from zero.
   What is plotted here is the offset l.i.= li(x) - li(2),
   often written as Li(x).
   (source: image by User:Dcoetzee at wikimedia)

It is extremely natural to infer that pi(x) < li(x), at any rate for large x,
and Gauss and other mathematicians commented on the high probability of this
conjecture.

The conjecture is not only plausible but is supported by all the evidence of
the facts. The primes are known up to 10,000,000, and their number at
intervals up to 1,000,000,000, and (1.34) is true for every value of x for
which data exist.

In 1912 Littlewood proved that the conjecture is false, and that there are an
infinity of values of x for which the sign of inequality in (1.34) must be
reversed. In particular, there is a number X such that (1.34) is false for
some x less than X. Littlewood proved the existence of X, but his method did
not give any particular value, and it is only very recently that an
admissible value, viz.

		X = 10^(10^(1034)

was found by Skewes. I think that this is the largest number which has
ever served any definite purpose in mathematics.

The number of protons in the universe is about 1080.  The number of
possible games of chess is much larger, perhaps 10^(1050) (in any case a
second-order exponential).

If the universe were the chessboard, the protons the chessmen, and any
interchange in the position of two protons a move, then the number of
possible games would be something like the Skewes number. However much the
number may be reduced by refinements on Skewes's argument, it does not seem
at all likely that we shall ever know a single instance of the truth of
Littlewood's theorem.

[I presume modern sources use the offset logarithmic integral primarily
because now it is indeed a lower bound.]

Contents

   I. The Indian mathematician Ramanujan			1
  II. Ramanujan and the theory of prime numbers 		22
 III. Round numbers						48
  IV. Some more problems of the analytic theory of numbers	58
   V. A lattice-point problem					67
  VI. Ramanujan's work on partitions				83
 VII. Hypergeometric series					101
VIII. Asymptotic theory of partitions				113
  IX. The representation of numbers as sums of squares 	132
   X. Ramanujan's function r(n) 				161
  XI. Definite integrals					186
 XII. Elliptic and modular functions				212
Bibliography							231