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CONTENTS
OF THIS SECTION
Last updated
13/06/07

C.J.Eliezer on Ramanujan
G.H.Hardy on Ramanujan
Krishnaswami Alladi - Book Review of Man who Knew Infinity
Ramanujan’s “Lost Notebook” Astounds Americans - 18 February 2005

Offsite Links

Indians of 20th Century - Srinivasa Ramanujan
Srinivasa Aiyangar Ramanujan - A Biography 
Srinivava Aiyangar Ramanujan - Slides
Ramanujan Journal

Books

*Srinivasa Ramanujan : A Mathematical Genius - K. Srinivasa Rao
*The Man Who Knew Infinity: A Life of the Genius Ramanujan - Robert Kanigel
*Ramanujan : Twelve Lectures on Subjects Suggested by His Life and Work - G. H. Hardy
*Ramanujan : Letters and Commentary (History of Mathematics, Vol 9)
- Bruce C. Berndt, Robert A. Rankin
The Lost Notebook and Other Unpublished Papers - Srinivasa Ramanujan
Ramanujan's Notebooks, Part 1 - Bruce C Berndt (Editor) 
Ramanujan's Notebooks, Part II - Bruce C. Berndt (Editor) 
Ramanujan's Notebooks, Part IV -  Bruce C. Berndt 
Chapter 16 of Ramanujan's Second Notebook Theta Functions and Q-Series (Memoirs of the American Mathematics Society, No. 315) -  C. Adiga, et al
Number Theory, Madras 1987 : Proceedings of the International Ramanujan Centenary Conference Held at Anna University, Madras,1987)
Number Theory : Ramanujan Mathematical Society January 3-6, 1996 Tiruchirapalli, India (Contemporary Mathematics, Vol 210)
- Vijaya Kumar Murty (Editor), et al

One Hundred Tamils of the 20th Century

Srinivasa Ramanujan

* indicates link to Amazon.com online bookshop

from a stamp issued by India to commemorate
the 75th anniversary of Ramunujan's birth

Srinivasa Aiyangar Ramanujan was born on 22 December 1887 in Erode, Tamil Nadu and died at the early age of 33 on 26 April 1920 in Kumbakonam. Ramanujan was one of the world's great mathematicians. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series.

Ramanujan, was largely self-taught in mathematics. He was given a fellowship to the University of Madras in 1903 but the following year he lost it because he devoted all his time to mathematics and neglected his other subjects. He married in 1909.

After publication of a brilliant research paper on Jacob Bernoulli's numbers in 1911 in the Journal of the Indian Mathematical Society his work gained recognition.

His papers were sent to several English mathematicians and Professor G H Hardy in Trinity College, Cambridge was quick to recognise Raminujan's genius. In 1914 Hardy invited Ramanujan to Trinity College, Cambridge, to begin an extraordinary collaboration. He sailed from India on 17 March 1914.

Ramanujan worked out the Riemann series, the elliptic integrals, hyper geometric series and functional equations of the zeta function. Ramanujan independently discovered the results of Gauss, Kummer and others on hyper geometric series. Ramanujan's own work on partial sums and products of hyper geometric series have led to major development in the topic. He was admitted as a Fellow of the Royal Society, though a Fellowship at Trinity College, Cambridge somehow eluded him.

He fell ill in 1917 and spent some time in nursing homes. After a slight improvement in his health, he sailed to India on 27 February 1919 and died there the following year.

Ramanujan left a number of unpublished notebooks filled with theorems that mathematicians have continued to study. Hardy passed on to G.N.Watson, Mason Professor of Pure Mathematics at Birmingham, the large number of manuscripts of Ramanujan that he had, both written before 1914 and some written in Ramanujan's last year in India before his death.

Watson published 14 papers under the general title 'Theorems stated by Ramanujan' and in all published nearly 30 papers which were inspired by Ramanujan's work.


C.J.Eliezer on Ramanujan the Mathematician

[see also C.J.Eliezer - One Hundred Tamils of 20th Century]

from a presentation at the First International Tamil Conference
1966 - Kuala Lumpur, Malaysia

The history of mathematics bears the imprint of many personalities of genius - of intellectual giants with unusual abilities and uncanny insights. Archimedes and Pythagoras. Newton and Lagrange. Descartes and Gauss, Euler and Einstein—were among the greatest of men. Intellectual passion and romance. discoveries and great moments of history have been associated with their names.

Among the exciting and unusual figures among the great creators of mathematics is Srinivasa Ramanujan, whose short life (1887 - 1920) had the stamp of genius and greatness though perhaps mingled with sadness and tragedy.

As one reflects on the life Ramanujan a thought that comes to mind is the possibility that his capabilities might well have gone unrecognised and opportunity may not have come his way for the growth and development of his talents. The help and encouragement of various men of influence who were themselves lovers of mathematics  played an important role in the early days of frustration and uncertainty.

One such person was Diwan Bahadur Ramachandra Rao, Collector at Nellore, who recalled in the following words his first interview with Ramanujan in 1910, when the latter was a twenty-two year old youngster. (He had by then left College. having failed in English in the First in Arts examination. lost his scholarship and was without a job)

"A nephew of mine perfectly innocent of mathematical knowledge said to me: ‘Uncle, I have a visitor who talks of mathematics: I do not understand him; can you see if there is anything in his talk’. And in the plenitude of my mathematical wisdom, I condescended to permit Ramanujan to walk into my presence A short figure . . . with one conspicuous feature—shining eyes ---- walked in with a frayed notebook under his arm. He was miserably poor . He had run away from Kumabakonam to get leisure in Madras to pursue his studies...

He opened his hook and began to explain some of' his discoveries. I saw quite at once that there was something out of the way: but my knowledge did not permit me to judge whether he talked sense or nonsense. Suspending judgement I asked him to come over again, and he did. And then he had gauged my ignorance and showed me some of his simpler results. These transcended existing books and I had no doubt that he was a remarkable man. Then, step by step, he led me to elliptic integrals and hyper geometric series and at last to his theory of divergent series not yet announced to the world converted me "

The reaction of Mr. Rao was typical of the admiration which lovers of mathematics gradually come to feel for Ramanujan's abilities and aroused their desire to make it possible for hint to continue with his mathematical pursuits. It would seem that there was in his environment an intellectual climate of appreciation of mathematical science, and a spontaneous affection and admiration for those who showed special mathematical talents. Such an emphasis reflects a traditional feature of Tamil society.

Among the first couplets of poetry which generations of children lisped were from the beloved and wise Avvaiyar:

number1.gif (2193 bytes)

And earlier in the second century Thiruvalluvar had said in Kura1 392

number2.gif (3222 bytes)

which may be rendered in English:

Numbers and letters - these are the twin eyes of the mind.

The inventionof the number zero in India, and the coming into use there of the present number system had accelerated the progress of number studies. And, over the centuries a love for mathematics, expressed in songs and poems and mathematical riddles (such as those which Thennali Raman had enunciated), had been nurtured in that society.

The phenomenon of Ramanujan gradually came to the notice also of certain British civil servants in India. These were mostly university men with a background of liberal education. Their support and enthusiasm were to play an important role on Ramanujan's future. While Ramanujan was working as an office clerk in the Madras Port Authority, his mathematical talents were brought to the attention of the Chairman of this body. This Chairman took advantage of the visit to Madras of Dr. T. G. Walker, F.R.S., Director-General of Observatories, Simla, to show him some of Ramanujan’s mathematical works. The result was a persuasive letter from Dr. Walker to the University of Madras in the following terms:

" . . . the character of the work that I saw impressed me as comparable in originality with that of a mathematical fellow in a Cambridge college ... the university would be justified in enabling S. Ramanujan for a few years to spend the whole of his time on Mathematics. without anxiety, as to his livelihood."

Dr. Walker as a former fellow of Trinity College. Cambridge, had known the value of the system of research fellowships at Cambridge where persons of promise are enabled to stay on at the university and devote their full time to research, without other dukes being required from them. The existence of such fellowships is today a valuable feature in universities which place emphasis on research and new learning.

The University of Madras went out of their normal provisions and awarded a special scholarship to Ramanujan thus enabling him to give up his clerkship and devote his full time to mathematics.

In the meantime, at the suggestion of a former teacher, Ramanujan began a correspondence with G. H. Hardy at Trinity College, Cambridge. Hardy later became for many decades the acknowledged leader of British mathematics. In his first letter Ramanujan wrote:

" . . . I had no university education. but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at Mathematics... Very recently I came across a tract published by you styled "Orders of Infinity`', in page 36 of which I find a statement that no definite expression has been as yet found for the number of prime numbers less than any given number. I have found an expression which very nearly approximates to the real result.... I would request you to go through the enclosed papers ...."

The papers enclosed contained enunciations of a hundred or more mathematical theorems. Hardy was amazed. He replied promptly in friendly and encouraging terms. He further urged that Ramanujan should enter Cambridge where with contact with many mathematicians his talents would find ample scope for creative mathematics. At first, Ramanujan would not entertain the idea of going abroad owing to his rigid observances of caste. Early in 1914 when a fellow of Trinity E.H. Neville. was invited by the University of Madras to visit and deliver some mathematics courses he was given by Hardy the task of persuading Ramanujan to go to Cambridge. This Neville did successfully. Neville further proposed to the University of Madras to award Ramanujan a scholarship to proceed to Cambridge:

'The discovery of the genius of' S. Ramanujan of Madras promises to be the most interesting event of our time in the mathematical world... I see no reason to doubt that Ramanujan himself will respond fully to the stimulus which contact with western mathematicians of the highest class will afford him. In that case his name will become one of the greatest in the history of mathematics and the University and the City of Madras will be proud to have assisted in his passage from obscurity to fame.''

The university responded and Ramanujan reached Cambridge in April 1914 in the company of Neville. There Hardy and Littlewood introduced him to new branches of mathematics, helped him to present his work in the form of papers suitable for publication in mathematical journals, and under their guidance his mathematical talents developed rapidly.

Hardy has commented on the difficulties of creating in Ramanujan the attitude of rigour characteristic of modern mathematics

'The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations. and theorems of complex multiplication to orders unheard of, whose mastery of continued fractions was. on the formal side at any rate beyond that of any mathematician in the world, who had found for himself the functional equation of the Zeta-function, and the dominant terms in many of the most famous problems in the analytic theory of numbers; and he had never heard of a doubly periodic function or of Cauchy's theorem, and had indeed but the vaguest idea of what a function of a complex variable was. His ideas of what constituted a mathematical proof were of the most shadowy description. All his results, new or old, right or wrong, had been arrived at by a process of mingled argument, intuition and induction...

So I had to try to teach him. and in a measure I succeeded. Though obviously I learnt from him much more than he learnt from me. His flow of original ideas showed no symptom of abatement "

Again:

"Of his extraordinary gifts there can be no question; in some ways he is the most remarkable mathematician I have ever known"

Ramanujan's uncanny intuition was his special asset Many of his results were so complicated that expert mathematicians had to put in great effort to provide acceptable proofs, and there still remain unproved results. Even in his younger days, on rising from bed. he would frequently jot down in his famous note book some formula which he would proceed to verify, though he was unable to supply a proof. He would say that the goddess of Nammakkal had inspired him with the formula during his sleep It would be of interest to speculate if his intuition would have remained as fertile if he had gone through a proper mathematical discipline of training. It could well be that systematic education sometimes tends to limit the intuitive capacities of children. In these days of mass education, where children are taught in large groups, the need for teachers to provide for the special talents of each child, especially in the case of mathematical talents, is important and worth greater attention than is given at present.

One of the happy features of Ramanujan's life at Cambridge had been the affection and regard in which Hardy, Littlewood, Neville and other mathematicians held him, reminding us of the international fraternity of mathematicians, who irrespective of origins of race, background and upbringing, are held together by a strong emotional basic loyalty which transcends the many differences.

Professor D. E. Daykin, my colleague in the University of Malaya tells me that a few years ago he attended a lecture by Prof. Neville on the life of Ramanujan. During most of the lecture, tears streamed down the cheeks of Prof. Neville. Among other incidents, Neville referred to the occasion when Ramanujan had borrowed some rare book from him and accidentally some water had poured on the book. Ramanujan was most unhappy, had written to several dealers in rare books, and succeeded in obtaining another copy. and with much apology he came to Neville and gave him the two copies, with an explanation of what had happened.

Ramanujan was in England for a little less than five years. Parts of the last two years were alas, in sanatoria, to arrest a tubercular tendency. There were periods of improvement, convalescence and relapse. But the mathematical activity continued. His work began to receive public recognition. He was elected to the Fellowship of the Royal Society.

On one occasion when Hardy visited him in hospital. the conversation casually went to the subject of the number of the taxicab in which he had travelled. Hardy remarked that the number was 1729 = 7 x 13 x 19 and thought that it was a dull number. Ramanujan however said that it was a very interesting number, being the smallest number expressible as a sum of two cubes in two different ways. So intimate was Ramanujan's knowledge about the different numbers. Littlewood once remarked that to Ramanujan every integer was a personal friend.

The climate of England was not helping his recovery, and his colleagues sadly planned his return to India. It was anticipated that he would fully recover in India's climate. The University of Madras made various financial provisions. and was contemplating the creation of a special Professorship for him. He was disturbed that his illness was preventing him from doing enough mathematics to justify his scholarship. He wrote to the University of Madras:

"I feel, however. that after my return to India, which I expect to happen as soon as arrangements can be made the total amount of money to which I shall be entitled will be much more than I shall require and I should hope that after my expenses in England have been paid, £50 a year will be paid to my parents and that the surplus after my necessary expenses are met should be used for some educational purpose such in particular as the reduction of school fees for poor boys and orphans and provision of books in school.. .

I feel very sorry that. as I  have not been well, I have not been able to do much mathematics during the last two years as before. I hope that I shall soon he able to do more and will certainly do my best to deserve the help that has been given me."

Ramanujan returned to Madras in April 1919. Respite best medical attention and the loving care of friends, his health did not improve. He died in April 20, at the age of 33, remaining mathematically active right to the end with a serenity of mind 'that held acquaintance with the stars undisturbed by space and time.'


References

Biography in Dictionary of Scientific Biography (New York 1970-1990).
Biography in Encyclopaedia Britannica.
G H Hardy, Ramanujan (Cambridge, 1940).
R Kanigel, The man who knew infinity : A life of the genius Ramanujan (New York, 1991).
S Ram, Srinivasa Ramanujan (New Delhi, 1979).
S R Ranganathan, Ramanujan : the man and the mathematician (London, 1967).
B Berndt, Srinivasa Ramanujan, The American Scholar 58 (1989), 234-244.
B Berndt and S Bhargava, Ramanujan - For lowbrows, Amer. Math. Monthly 100 (1993), 644-656.
J M Borwein and P B Borwein, Ramanujan and pi, Scientific American 258 (2) (1988), 66-73.
L Debnath, Srinivasa Ramanujan (1887-1920) : a centennial tribute, International journal of mathematical education in science and technology 18 (1987), 821-861.
G H Hardy, The Indian mathematician Ramanujan, Amer. Math. Monthly 44 (3) (1937), 137-155.
C T Rajagopal, Stray thoughts on Srinivasa Ramanujan, Math. Teacher (India) 11A (1975), 119-122, and 12 (1976), 138-139.
R A Rankin, Ramanujan's manuscripts and notebooks, Bull. London Math. Soc. 14 (1982), 81-97.
R A Rankin, Ramanujan's manuscripts and notebooks II, Bull. London Math. Soc. 21 (1989), 351-365.
R A Rankin, Srinivasa Ramanujan (1887- 1920), International journal of mathematical education in science and technology 18 (1987), 861.
R A Rankin, Ramanujan as a patient, Proc. Indian Ac. Sci. 93 (1984), 79-100


G.H.Hardy on Ramunujan

Godfrey Hardy was the Cambridge mathematician who `discovered' the great Indian mathematician Ramanujan. This is a condensed version of the first chapter in *Ramanujan : Twelve Lectures on Subjects Suggested by His Life and Work  by G. H. Hardy (The condensation is courtesy: Zimath) 

Introduction

I have set myself a task that is genuinely difficult, even impossible --- to form some sort of reasoned estimate of 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 judgement only, that he was in some sense a very great mathematician.

The difficulties in judging Ramanujan are clear --- he was an Indian, I am an Englishman, and the two parties have always found it hard to understand one another. He was at best, a half-educated Indian, since he never could rise to be even a "failed B.A.". He worked for most of his life ignorant of modern European maths, and died when he was thirty and when his mathematical education had in some ways hardly begun. He published abundantly (at least 400 pages worth) but left behind even more unpublished stuff. While this work includes much that is new, about two-thirds is rediscovery, that too usually imperfect rediscovery.

His early life

Srinivasa Aiyangar Ramanujan was born in 1887 in a poor Brahmin family at Erode near Kumbakonam, a fair sized town in the Tanjore district of Tamil Nadu. His father was a clerk in a cloth-merchant's office in Kumbakonam. He was sent at seven to the local high school and stayed there nine years. By the time he was in his early teens it was common knowledge that he was more than just a brilliant student, discovering for instance the relationship between circular and exponential functions (cos a + i sin a = e^ia). This of course had been discovered by Euler before, as he found out much to his chagrin later on.

When he was sixteen he came across "A synopsis of elementary results (actually, over 6000 theorems) in pure and applied mathematics" by George Carr, . This enthusiastic book served to introduce Ramanujan to the real world of mathematics, but in a highly personal style that relegated the proofs to mere footnotes. Ramanujan went through the entire book methodically and excitedly, proving its theorems by himself, often as he got up in the morn. He claimed that the goddess of Namakkal inspired him with formulae in dreams.

His religion

Was he religious? Certainly he observed his duties as a high-caste Hindu assiduously, like being a faultless vegetarian and cooking all his food himself (after changing into his pyjamas first). And while his excellent Indian biographers (Seshu Aiyar and Ramachandra Rao) say he believed in the existence of a Supreme Being, in Kharma, Nirvana and other Hindu tenets, I suspect he was not affected by religion any more than as a collection of rules to be followed. He told me once, to my surprise, that all religions seemed to him to be more or less equally true.

Some thought, and may still think, of Ramanujan as a unintelligible manifestation of the mystic East. Far from it! He had his oddities, no doubt mostly originating from his different culture, but he was as reasonable, sane and shrewd as anyone I've met. He was a man in whom society could take pleasure, with whom one could sip tea and discuss politics or mathematics. He was a normal human being who happened to be a great mathematician.

The rest of his life

Back to his early days. Thanks to his fine academic school record, he won a scholarship to university. But there he spent his time doing mathematics at the expense of his other subjects, which he consequently failed. His scholarship was not renewed. Further attempts to complete his degree failed. He married at 22 but could not find a university post, despite the fervent attempts of some influential Indians he had impressed with his results, Ramaswami Aiyar and his two biographers. Finally (at 25) in 1912 he found his first real job, a mundane clerical one in the Port Trust of Madras. But the damage had been done --- the years between 18 and 25 are the critical ones in a mathematician's life and his genius never again had the chance of full development. This, and not his early death, was the real tragedy, that his genius was misdirected, sidetracked and to some extent distorted by an inelastic and inefficient educational system.

But the foundations of at least a partial recovery had been laid. In 1911 he had published his first substantial paper and the following year two Britons, Sir Gilbert Walker and Sir Francis Spring secured for him a special scholarship (60 pounds a year) that was enough for a married man to live in tolerable comfort. He wrote to me in early 1913, and Professor Neville and myself got him to Britain after much difficulty in 1914. He then had three years of continuous work before falling ill in mid-1917. He was only able to work spasmodically (but as well as ever) after this, and died in 1920.

His letters to me

The stories, true and false, of what happened when I read the letters of an unknown Hindu clerk have been well spread --- like how I first stored them in my wastepaper basket before retrieving them for a second look, and so on. His letters contained the bare statement of about 120 theorems. Several of them were known already, others were not. Of these, some I could prove (after harder work than I had expected) while others fairly blew me away. I had never seen the like! Only a mathematician of the highest class could have written them. They had to be true, for if they were not, no one would have the imagination to invent them. A few were definitely wrong. But that only added credence to my feeling that the writer was totally honest, since great mathematicians are commoner than frauds of the incredible skill that would be needed to create such a letter.

My collaboration with him

While his mind had been hardened by the time I had access to him, Ramanujan could still learn new things, and learn them well. It was impossible to teach him systematically, but he gradually absorbed new points of view (like why proofs were important!). But there were theorems he should have revelled in, but never used, nor ever seemed to need! The line between what he learnt from books and learnt for himself was always very hazy. And here I shall have to apologize to the world for not asking him about such matters. For I could have easily asked him, seeing him daily, and he would have been perfectly willing to tell me. But I had no idea he was going to die so soon, and it seemed ridiculous to worry about how he had found this or that theorem when he was showing me half a dozen new ones almost every day.

How good was he?

In his favourite topics, like infinite series and continued fractions, he had no equal this century. His insight into algebraic formulae, often (and unusually) brought about by considering numerical examples, was truly amazing. But in analytic number theory, a subject he is often associated with, I do not believe he actually knew that much. He certainly contributed little of significance that was not known already. And in a subject that relied so much on proof, a subject where intuition had a bad habit of coming unstuck, he produced much that was false.

I have in the past tried to say things like "his failure was more wonderful than any of his triumphs", but that is absurd. It is no use trying to pretend that failure is something else. All we can say is that his failures give us additional, surprising evidence of his imagination and versatility. And we can respect him as one who let his mind run free, instead of keeping it under saddle and blinkers like so many others do.

Conclusion

But the reputation of a mathematician cannot be made by failures or by rediscoveries; it must rest primarily, and rightly, on actual and original achievement. And it is still possible to justify Ramanujan on these grounds.


 Book Review by Krishnaswami Alladi
*
The Man Who Knew Infinity: A Life of the Genius Ramanujan - Robert Kanigel, 438pp. Charles Scribners 1991

The life story of Srinivasa Ramanujan (1887-1920), the legendary mathematician from India, is astonishing in many ways. Without any formal training he produced results of incredible depth and beauty that challenged some of the finest minds in England during the beginning of this century. What is even more remarkable is that he discovered many of these results while in South India, where the traditional Hindu way of life had changed little over the centuries. professor G H Hardy of Cambridge University was immensely impressed by these discoveries and arranged for Ramanujan to go to England. Although Ramanujan had great difficulties in adjusting to the British way of life, he wrote several fundamental papers in Cambridge. But the rigors of life in England during World War 1, combined with his own peculiar habits, led to a rapid decline in his health. This forced him to return to India, where he died a year later.

Even from his deathbed Ramanujan made startling discoveries on mock theta functions, about which I shall comment later. During his all-too-brief life of 32 years, he left numerous results that were unique in their beauty, deep, fundamental and of lasting value; together they place him among the greatest mathematicians in history. Hardy published 12 elegant lectures in 1940 explaining many facets of Ramanujan's work; in the past few decades, the American mathematicians George Andrews, Richard Askey and Bruce Berndt have published several books and monographs expanding on Ramanujan's ideas. But this is the first detailed biography of Ramanujan, and Robert Kanigel is successful in bringing out Ramanujan's drama in a most interesting manner.
Ramanujan was born on December 22, 1887, in an orthodox South India Brahmin family. His parents, who lived in Kumbakonam, a small town in what is now the state of Tamil Nadu, had been childless for many years. They prayed to the Goddess Namagiri, the deity in the neighbouring town of Nammakkal, to bless them with a child. Promptly Ramanujan's mother became pregnant and gave birth in the nearby town of Erode, her mother's place. To his parents Ramanujan was a divine gift, and the goddess of Namakkal was held in great veneration by his family. Ramanujan grew up in a traditional Hindu environment, learning stories from the great epics and verses from the Hindu holy scriptures.

Ramanujan showed signs of his special mathematical talent early. He kept notebooks in which he would regularly jot down his findings. What was most strange was the manner in which he arrived at his results, and this still remains a mystery. Often, he would suddenly get up in the middle of the night and immediately write down identities involving infinite series and products. Those near him have said that Ramanujan used to mention that the Goddess of Namakkal appeared to him in his dreams and presented him with these incredible formulae. As an agnostic, Ramanujan's mentor, Hardy, dismissed the story of the goddess as mere fable. That such divine inspiration is often considered to be the cause of work of exceptionally high quality, however, is more acceptable to a Hindu than to someone steeped in Western tradition. For instance, Hindus believe that it was the blessing of Goddess Kali that instantly transformed Kalidasa from a shepherd to a poet par excellence!

During his school years, Ramanujan's excessive preoccupation with mathematics led to his neglect of other subjects, and he had to drop out of college. In 1909, with the intention of making him more responsible, Ramanujan's mother arranged to have him married to Janaki, who was then only a nine-year-old girl.

Next, Kanigel describes Ramanujan's efforts in approaching influential people for financial assistance so that he could pursue his research unhindered by the distractions of a job. What Ramanujan really needed was the attention of a leading mathematician. India was a British colony, and so it was natural for him to write letters to British professors stating his results. And it was Hardy who responded favourably.

G H Hardy was, at the beginning of this century, leading the revival of British mathematicians, which had taken a back seat in the post-Newtonian era. Because he was well-known, Hardy was used to receiving letters from amateur mathematicians who made false claims about the solutions of famous problems. So in January 1913, when a letter from Ramanujan arrived containing a long list of formulae without any proofs, Hardy's first reaction was to ignore the letter as one written by a fraud. However, a closer look showed that there were several beautiful formulae, some of which defeated him completely. Hardy cam to the conclusion that it was more probable that Ramanujan was a genius, because a fraud would not have had the imagination to invent such identities! A correspondence followed, and Hardy invited Ramanujan to Cambridge so that his raw, untutored genius could be given a sense of direction.

Both Ramanujan's reaction to this invitation and his mother's were negative. At that time, orthodox Hindus believed that it was sinful to cross the seas. Once again the Goddess of Namakkal provided the solution! This time his mother had a dream in which she saw Ramanujan being honoured in an assembly of European mathematicians, and the goddess instructed her not to stand in the way of her son's recognition. So, finally, Ramanujan sailed for England in 1914. Ramanujan's reluctance to go to England without his mother's permission was a typical Hindu reaction. It is a common practice in India, even today, to seek the blessings of elders before embarking on a voyage.

During his few years in England, the rise of Ramanujan's reputation was meteoric. In each of his frequent discussions with Hardy, he showed several new results. For example, under Ramanujan's magical hand, the theory of partitians that had been founded by Euler underwent a glorious transformation. Ramanujan discovered several astonishing new theorems on partitions involving congruences and continued fractions. In collaboration with Hardy, he showed how to obtain an accurate formula for the number of partitions of an integer. This is the famous circle method so widely used in number theory today. In another paper with Hardy, he began the investigation of round numbers that led to the creation of probalistic number theory several years later by mathematicians such as Paul Erdos. His research was so impressive that he was elected a Fellow of The Royal Society (F R S) in 1918.

Ramanujan, who was a gregarious and orthodox Brahmin, found himself in an awkward position amid educated Englishmen who were aloof. Socially, Hardy was the opposite of Ramanujan. This book is a dual biography, of Ramanujan and Hardy, and Kanigel succeeds wonderfully in showing the gulf that separated the two. What bridged this gap was mathematics, but here too they differed considerably in the way they thought. Ramanujan was a genius who conjectured and made giant leaps of imagination; as a seasoned mathematician, Hardy put emphasis on rigor and proceeded by logical step-by-step reasoning.

England's climate proved disastrous for Ramanujan's health. He never adjusted to the cold weather. He was in and out of sanatoriums, being treated mainly for tuberculosis. Having been used to the curries and spices of India, he found English to be tasteless. In 1919 his health became so bad that he returned to India. He died the following April in Madras.

Hardy felt that the real tragedy was not Ramanujan's early death, but the fact that he had wasted much time in India rediscovering past work. He argued that the best creative work is done when one is very young and, therefore, that at the time of his death, Ramanujan was perhaps already pas his prime. But here Hardy may have been wrong. Ramanujan's now-famous work on the mock theta functions was done during his last few months in India. He wrote one last letter to Hardy summarizing his discoveries. They are now considered to be among his deepest contributions. Ramanujan was definitely on the rise, and he could have reached even greater heights had he lived longer.

Hardy compared Ramanujan to Euler and Jacobi as a genius. Yet he was of the opinion that Ramanujan's work was strange, and that it lacked the simplicity of the very greatest works. With recent advances in the theory of modular forms and the research of Andrews on Ramanujan's "Lost Notebook", we now realize that Ramanujan's work is more fundamental than Hardy had ever imagined. Ramanujan's equations are now being used to compute pi (the ratio of the circumference of a circle to its diameter) to a billion digits! Atle Selberg of the Institute for Advanced Study at Princeton has said that it will take many more decades, possibly more than a century, to fully understand Ramanujan's contributions.

The fascinating story of the discovery of the "Lost Notebook" is described in this book. Shortly after Ramanujan's death, his widow, Janaki, collected all the loose sheets on which Ramanujan had scribbled mathematics and sent them to hardy. They contained more than 600 formulae, including many on mock theta functions. Hardy handed this manuscript to G N Watson, who wrote two papers on this topic. After Watson's death this manuscript was placed along with Watson's papers at the Wren Library in Cambridge University, and the mathematical world remained unaware of its significance. In 1976, however, Andrews stumbled across the manuscript while doing some reference work at Cambridge University. He recognized it instantly as a priceless treasure and has been analyzing its contents ever since.

When Ramanujan's centenary was celebrated in India in december 1987, mathematicians from all over the world came to pay homage to this legendary genius. There were several conferences held in India, of which two were in Madras. The first of these was at Anna University, for which Andrews had come in connection with a session on number theory that I organized. Mrs Ramanujan, who was 87 years old, was present on the opening day. Kanigel says "Andrews, his voice choked with emotion, presented Janaki with a shawl. It was she who deserved the credit for the Lost Notebook, he said, since it was she who kept it together while Ramanujan lay dying." At the second conference, India's prime minister, Rajiv Gandhi, presented two copies of the "Lost Notebook", the first one to Mrs Ramanujan and the other to George Andrews.

Bruce Berndt is editing the notebooks of Ramanujan. He has published three volumes, and two more are forthcoming. Andrews and Berndt have plans to edit the "Lost Notebook". Owing to the efforts of Andrews, Askey and Berndt, it is now possible to include Ramanujan's work as part of the regular graduate mathematics curriculum. And, by reading this fascinating biography, students will be drawn to a study of Ramanujan's spectacular results.


Ramanujan’s “Lost Notebook” Astounds Americans
18 February 2005

Death snatched India’s Srinivasa Ramanujan when his genius had just blossomed - at age 32 in 1920. Eighty-five years later, it’s still taking several expert mathematicians in America a lifetime to decipher just a portion of his incandescent genius. Remarkably too, some of that work is getting financial support from the National Security Agency.

Just consider what’s happening now. Prof. George Andrews of Pennsylvania State University, one of the world's most eminent mathematicians, is conducting a series of seminars from January 25 until March 22, 2005 at the University of Florida. Andrews, who has spent the past 30 years studying Ramanujan’s considerable output, is providing insights into ‘Number Theory and Combinatorics’ through six lectures on topics related to Ramanujan's ‘Lost Notebook.’ The lost notebook arises from the last year of Ramanujan's life and contains approximately 650 assertions without proofs.

According to Prof. Krishnaswami Alladi, Chairman of the Department of Mathematics at the University of Florida, who recently visited Ramanujan’s birthplace, along with Prof Andrews, the lectures will seek to unveil ‘what did Ramanujan have up his sleeve?’ Alladi’s research is in Number Theory, an area where Ramanujan has made spectacular contributions. He is also the Editor-in-Chief of The Ramanujan Journal, an international publication devoted to all areas of mathematics influenced by Ramanujan.

In his introduction to the first seminar on January 15th, Andrews described the life of Ramanujan, the discovery of his Lost Notebook, and attempted to describe some of his surprising achievements. The third lecture was delivered on February 15. Three more are scheduled.

Essentially, Ramanujan's legacy consists of 4,000 formulas on 400 pages filling 3 volumes of notes, all densely packed with theorems of incredible power but without any commentary or, which is more frustrating, any proof. In 1976, however, a new discovery was made. One hundred and thirty pages of scrap paper, containing the output of the last year of his life, was discovered by accident in a box at Trinity College. This is now called Ramanujan's "Lost Notebook."

Commenting on the Lost Notebook, mathematician Richard Askey says, "The work of that one year, while he was dying, was the equivalent of a lifetime of work for a very great mathematician. What he accomplished was unbelievable. If it were a novel, nobody would believe it." To underscore the difficulty of the arduous task of deciphering the "notebooks," mathematician Jonathan Borwein and Peter Borwein have commented, "To our knowledge no mathematical redaction of this scope or difficulty has ever been attempted."

Prof. Andrews shot to fame in the 1970s when he discovered Ramanujan's Lost Notebook at the Wren Library in Cambridge University and wrote a series of important papers in Advances in Mathematics in which he explained Ramanujan's spectacular results in the context of current research, and in that process made fundamental improvements as well.

"There is still much to understand about the implications of many results in the Lost Notebook and their connections with current research which is one of the reasons to edit the Lost Notebook," said Professor Andrews. The first of these volumes will appear in 2005 and at least two more volumes will be forthcoming. "The mathematical content of the Lost Notebook is so immense, that it is difficult to predict at this time how many volumes it will take to completely edit it," he added.

According to Prof. Alladi, during the 1987 Ramanujan Centennial, the printed form of Ramanujan's Lost Notebook by Springer-Narosa was released by Prime Minister Rajiv Gandhi, who presented the first copy to Janaki Ammal Ramanujan, the late widow of Srinivasa Ramanujan, and the second copy to Professor Andrews in recognition of his contributions.

Ramanujan’s mathematical wizardry is also the reason why University of Illinois Prof. Bruce Berndt, an analytic number theorist with strong interests in several related areas of classical analysis, has devoted 31 years of his research to proving the claims left in three notebooks and a "lost notebook" by the Indian genius upon his death in 1920. Twenty-one students have completed doctoral theses under Berndt's direction, and currently, five Ph.D. students are writing their dissertations under his direction. Most are focusing on material in the lost notebook or on research inspired by Ramanujan.

The three original notebooks contain approximately 3300 results. The project of finding proofs for these claims took Berndt over twenty years to accomplish, and an account of this work can be found in his books, Ramanujan's Notebooks, Parts I-V, published by Springer--Verlag in the years 1985, 1989, 1991, 1994, and 1998. Also during this time, Berndt and Robert A. Rankin wrote Ramanujan Letters and Commentary and Ramanujan Essays and Surveys, both published jointly by the American and London Mathematical Societies in 2001.

Berndt’s research in this direction continues, as he and Andrews plan to publish volumes on Ramanujan's "lost" notebook, analogous to those published on the ordinary notebooks. They are currently "editing" Ramanujan's Lost Notebook, which will be published by Springer later this year.

When an interviewer for Frontline asked Brendt if he found any of Ramanujan’s results difficult to decipher, he admitted: “Oh yes. I get stuck all the time. At times I have no idea where these formulae are coming from. … There are times I would think of a formula over for about six months or even a year, not getting anywhere. Even now there are times when we wonder how Ramanujan was ever led to the formulae. There has to be some chain of reasoning to lead him to think that there might be a theorem there. But often n this is missing. To begin with, the formulae look strange but over time we understand where they fit in and how important they are than they were previously thought to be.”

Ramanujan's Passion

Born in India in 1887, Ramanujan was a mathematical genius whose work continues to surprise mathematicians into the 21st century. His work is filled with surprises. At the Ramanujan centenary conference at the University of Illinois, it was physicist Freeman Dyson who proclaimed, "That was the wonderful thing about Ramanujan. He discovered so much, and yet he left so much more in his garden for other people to discover."
Born into poverty, Ramanujan grew up in southern India, and although he had little formal training in mathematics, he became hooked on mathematics. He spent the years between 1903 and 1913 cramming notebooks with page after page of mathematical formulas and relationships that he had uncovered.

Ramanujan's life as a professional mathematician began in 1914 when he accepted an invitation from the prominent British mathematician G.H. Hardy to come to Cambridge University. He spent 5 years in England, publishing many papers and achieving international recognition for his mathematical research.

Though his work was cut short by a mysterious illness that brought him back to India for the final year of his life, Ramanujan's work has remained a subject of considerable interest.

The 600 formulae that Ramanujan jotted down on loose sheets of paper during the one year he was in India, after he returned from Cambridge, are the contents of the `Lost' Note Book found by Andrews in 1976. He was ailing throughout that one year after his return from England (March 1919 - April 26, 1920). The last and only letter he wrote to Prof. Hardy, from India, after his return, in Jan. 1920, four months before his demise, contained no news about his declining health but only information about his latest work: ``I discovered very interesting functions recently which I call `Mock' theta-functions. Unlike the `False' theta-functions (studied partially by Prof. Rogers in his interesting paper) they enter into mathematics as beautifully as ordinary theta-functions. I am sending you with this letter some examples ... ''.

The following observation of Richard Askey is noteworthy: ``Try to imagine the quality of Ramanujan's mind, one which drove him to work unceasingly while deathly ill, and one great enough to grow deeper while his body became weaker. I stand in awe of his accomplishments; understanding is beyond me. We would admire any mathematician whose life's work was half of what Ramanujan found in the last year of his life while he was dying''.

As for his place in the world of Mathematics, this is what Berndt says: `` Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself a score of 25, Littlewood 30, Hilbert 80 and Ramanujan 100''.

In 1957, with monetary assistance from Sir Dadabai Naoroji Trust, at the instance of Professors Homi J Bhabha and K. Chandrasekaran, the Tata institute of Fundamental Research published a facsimile edition of the Notebooks of Ramanujan in two volumes, with just an introductory para about them.

The formidable task of truly editing the Notebooks was taken up in right earnest by Berndt in May 1977 and his dedicated efforts for nearly two decades has resulted in the Ramanujan's Notebooks published by Springer-Verlag in five Parts, the first of which appeared in 1985.

Between 1903 and 1914, before Ramanujan went to Cambridge, he compiled 3,542 theorems in the notebooks. Most of the time Ramanujan provided only the results and not the proof. Berndt says: "This is perhaps because for him paper was unaffordable and so he worked on a slate and recorded the results in his notebooks without the proofs, and not because he got the results in a flash."

The three original Ramanujan Notebooks are with the Library of the University of Madras, some of the correspondence, papers/letters on or about Ramanujan are with the National Archives at New Delhi and the Tamil Nadu Archives, and a large number of his letters and connected papers/correspondence and notes are with the Wren Library of Trinity College, Cambridge. The Ramanujan Institute for Advanced Study in Mathematics of the University of Madras is situated at a short distance from the famed Marina Beach and is close to the Administrative Buildings of the University and its Library. Mrs. Janakiammal Ramanujan, the widow of Ramanujan, lived close to the University's Marina Campus and died on April 13, 1994. A bust of Ramanujan, sculpted by Paul Granlund was presented to her and is now with her adopted son Mr. W. Narayanan, living in Triplicane near Chennai.

Ramanujan's story has been told recently by mathematician Ian Stewart, writing in the magazine "New Scientist". Stewart brings out strongly the special nature of Ramanujan's genius, which is his amazing intuition of the correctness of a complicated mathematical result. He points out what a lot this has to do with Ramanujan's lack of a formal education. Because of this, many of Ramanujan's proofs have serious gaps in them. They do not follow step by step in the way that a conventional proof would. In fact, later mathematicians have often had to wrestle long and hard to produce proofs that are completely watertight: to dot all the i's, and cross all the t's, as it were. The incredible thing is that for all their lack of technical rigor, almost all of Ramanujan's results turned out to be correct.

Alladi says that his visit to Ramanaujam’s birthplace was a dream come true. “What an inspiration to see this small humble home from where so many significant mathematical discoveries poured forth ” says Alladi. He informs that Shanmugha Arts, Science, Technology and Research Academy (SASTRA), a private university whose main campus is located in the town of Tanjore, purchased the home of Ramanujan in 2003, and has since maintained it as a museum. Research chairs established at the SASTRA centre at Kumbakonam — two by the Department of Science and Technology, Government of India and one by City Union Bank Ltd. — encourage research in the field of mathematics in honor of Ramanujan.

Meanwhile the founding has been announced of the "Ramanujan Prize for Young Mathematicians from Developing Countries" by the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy, in cooperation with IMU, and with support from the Niels Henrik Abel Memorial Fund, Norway. The Prize will be awarded annually for the highest mathematical achievement by young researchers from developing countries, which conduct their research in a developing country. The recipient must be less than 45 years old. Work in any branch of the mathematical sciences is eligible for the prize. The Prize amount will be $10,000. The goal is to make the selection of the first Prizewinner in 2005.  

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