K. Srinivasa Rao

The Institute of Mathematical Sciences, Chennai 600 113

. (E-mail : rao@imsc.ernet.in)

Introduction

Srinivasa Ramanujan, hailed as one of the greatest mathematicians of this century, left behind an incredibly vast and formidable amount of original work, which has greatly influenced the development and growth of some of the best research work in mathematics of this century. He was born at Erode, on Dec. 22, 1887. There were no portents to indicate that he would, in a short life-span of 32 years 4 months and 4 days, become comparable to the all-time great Euler, Gauss and Jacobi, for natural genius.

There are two aspects of interest to biographers and mathematicians regardingRamanujan: his life and his work. Mathematicians, who are interested in his work, have to contend with not only his publications in journals which are precise and profound, but also with his Notebooks which are a treasure house of intriguing results stated without proofs and lacking perspective with contemporary mathematical work. Those who attempt to write biographic articles on Ramanujan have to surmount the time barrier to reconstruct a story from all the indirect information accessible and to them, Hardy on Ramanujan is akin to Boswell on Samuel Johnson. The challenge to the mathematicians who work on any of his thousands of recorded results, which are still shrouded in mystery, is to prove the same with what was accessible to Ramanujan in those days in the form of books and publications. While the individual writer’s perception of Ramanujan will depend upon his/her background and imagination, the task of the mathematician is perhaps unenviable, in comparison.

Anyone who ever heard of Srinivasa Ramanujan and reads the compelling rags-to-intellectual-riches story of Ramanujan contained in the two Notices, one by G.H. Hardy and the other by Dewan Bahadur R. Ramachandra Rao and P.V. Seshu Iyer, published in the Collected papers of Srinivasa Ramanujan , would be moved by the achievements of the unorthodox mathematical genius under adverse circumstances.

The lack of formal education, lack of appreciation and a job, in the beginning of his career and ill health during the last few years of his life, did not prevent him from being creative in Mathematics. This is indeed something not easy to comprehend, for often one would buckle under similar trying circumstances. In these lectures, I will present an account of his romantic life, provide a few glimpses into his mathematics and relate the increasing interest in his work and its relevance even today.

Formal education

Ramanujan’s father, Mr. K. Srinivasa Iyengar, was an accountant to a cloth merchant in Kumbakonam. His mother was Komalattammal and Erode was her parental home. He was the first of three sons to his parents. Very little is known about his father and not even a photograph of his seems to be available. His mother was convinced of the greatness of Ramanujan and she zealously protected and projected his interests all through his life. She is portrayed as a shrewd, cultured lady and her photograph is available in some books on Ramanujan.

Ramanujan was sent to Kangeyam Primary School in Kumbakonam at the age of seven. During his school days, he impressed his classmates, senior students and teachers with his extraordinary intuition and astounding proficiency in several branches of mathematics - viz. arithmetic, algebra, geometry, number theory and trigonometry. In later years a friend of his, C.V. Rajagopalachari, recounted the following incident which happened when Ramanujan was in his third form: In an arithmetic class on division, the teacher said that if three bananas were given to three boys, each boy would get a banana. The teacher generalised this idea and said that any number divided by itself would give one.

Ramanujan asked:*Sir, if no banana is distributed to no student, will everyone still get a banana ?*

Another friend who took private tuition from Ramanujan also recalled that Ramanujan used to ask about the value of zero divided by zero and then answer that it can be anything since the zero of the denominator may be several times the zero of the numerator and vice versa and that the value cannot be determined. He stood first in the Tanjore District Primary Examinations held in November 1897, and this entitled him to a half-fee concession in the Town High School at Kumbakonam, where he studied from 1898 to 1903, until he passed the Matriculation Examination of the University of Madras (1904). At the age of 12, Ramanujan is said to have worked out the properties of arithmetical, geometrical and harmonic progressions. Once a senior school student , posed to Ramanujan, who was in the fourth year at school, the following problem:

If px + y = 7 and x + py = 11, what are the values of x and y ?

Ramanujan’s immediate reply to this questionn– which was expected to be tackled by only a sixth year student – that x = 9 and y = 4, won for him a friend who in later years took him to the collector of Nellore.

The senior mathematics teacher of the school, Ganapathy Subbier, had such confidence in Ramanujan’s ability that year after year he entrusted Ramanujan with the task of preparing a conflict free time-table for the school, which had about 1500 students and 30 or more teachers. Ramanujan won prizes for his outstanding performance in mathematics and mastered Loney’sTrigonometry, Part II, in his fourth year at school. He won many prizes in his second, fourth and sixth years at High School.

To augment the family income, Ramanujan’s mother took in a couple of students from Tirunelveli and Tiruchirapalli as boarders. Noticing Ramanujan’s precocity in mathematics these undergraduate students are purported to have given him an elementary introduction to all branches of mathematics. In 1903, through these friends from the Kumbakonam Government College, Ramanujan obtained G.S. Carr’s: A Synopsis of Elementary Results, a book on Pure Mathematics, which contained propositions, formulae and methods of analysis with abridged demonstrations, published in 1886.

Carr presented in this book 4865 formulae , without proofs, in algebra, trigonometry, analytical geometry and calculus. This book is similar to the modern day compilations like the Table of Integrals, Series, and Products, by I.S. Gradshteyn and I.M. Ryzhik (Academic Press, New York, 1994). Prof. P.V. Seshu Aiyar and Mr. R. Ramachandra Rao, in their biographies of Ramanujan state that:

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.

It is the considered opinion of many that in proving one formula, he discovered many others and thus, Ramanujan laid for himself a foundation for higher mathematics. Also, at about this time, he started noting his results in Notebooks. The first public recognition of his extraordinary prowess came when he was awarded a special prize – the Sri K. Ranganatha Rao Prize e– at the annual prize distribution ceremony of the Town High School, in 1904, for proficiency in mathematics. Ramanujan passed his Matriculation Examination in 1904 and joined the Government Arts College in Kumbakonam. As a result of his success in a competitive examination in Mathematics and English composition, he secured the Junior Subrahmanyam Scholarship. In the F.A. (First Examination in Arts) Class, Ramanujan had to study English, Sanskrit, Mathematics, Physiology and the History of Rome and Greece. Partly due to his pre-occupation with researches into mathematics, he neglected the study of other subjects. He went to his mathematics lecturer with a number of original and very ingenious results in finite and infinite series. Prof. P.V. Seshu Aiyar exhorted him but advised him not to neglect the study of other subjects. Unfortunately, he did not pass in English and Physiology and hence was not promoted to the senior F.A. class in January 1905. He lost his scholarship. His mother, who played a domineering role in his life, tried to persuade the Principal of the Government Arts College to take note of Ramanujan’s extraordinary mathematical ability and appealed for a continuance of the scholarship, but to no avail.

Ramanujan’s failure to get promoted to the senior F.A. class marked the beginning of a very trying period in his life. It is not clear what he did in 1905, when he discontinued his studies and spent some months in (the present day) Andhra Pradesh region, when he set out from Kumbakonam, for the first time. He joined Pachaiyappa’s College in Madras, in the F.A. class again, in 1906. One of his classmates, T. Devaraja Mudaliar, recalls that the Chief Professor of Mathematics, P. Singaravelu Mudaliar, considered an acquisition by Pachaiyappa’s College since he had the reputation of being a very successful teacher for the B.A. class, waited for Ramanujan’s assistance to solve difficult problems in mathematical journals. He also recalls that a junior mathematics teacher of the F.A. class, Prof. N. Ramanujachari, allowed Ramanujan to go to the board to show the solutions to the difficult problems in algebra or trigonometry using fewer steps than the ones used by him. Senior students of the B.A. Class also sought Ramanujan’s help in mathematics

Ramanujan who was a strict vegetarian should have abhorred the dissection of the frog in the Physiology classes. Once, to a question on the digestive system, he is supposed to have provided a skimpy answer which he concluded with : Sir, this is my undigested product of the Digestion chapter. Please excuse me. Another classmate of his at Pachaiyappas College recalls that Ramanujan rarely got more than 10 contempt and got something more, say 15 % to 20 % in Greek and Roman History, but managed to get about 25 % in English. However, Ramanujan considered the problems given in· · · textbooks in Geometry, Algebra, and Trigonometry to be mental sums.

In 1906, while studying at Pachaiyappa’s College, Ramanujan lived with his grandmother in a house in a lane in George Town, Madras. After about three months, Ramanujan fell ill and discontinued his studies. However, he appeared privately for the F.A. examination in 1907. Though he secured a centum in mathematics, he failed to secure pass marks in other subjects. This marked the end of his formal education.

Formative years :

It was during the period, 1907 - 12, that Ramanujan was frantically in search of a benefactor and started making contacts with those who could help him in his quest for a job to eke out a livelihood. He continued to stay in Madras after his formal education came to an end in 1907. According to Hardy:

The years between 18 and 25 are the critical years in a mathematician’s career. During his five unfortunate years (1907-1912) his genius was misdirected, side-tracked and to a certain extent distorted. (Hardy).

Despite the pecuniary circumstances and the stresses and strains of day-to-day existence, Ramanujan started noting down his mathematical results in Notebooks. By 1909, his Notebooks were precious to Ramanujan. For, one (F.A.) classmate of his, states that Ramanujan fell ill in 1909, while living in George Town, Madras, and on a Doctor’s advise, when he was being sent to the home of his parents in Kumbakonam, Ramanujan entrusted him with his Notebooks for safe keeping and stated:*If I die, please hand them over to Prof. Singaravelu Mudaliar or to the British Professor– Edward B Ross – Madras Christian College.*

Another college mate of Ramanujan has stated that during his collegiate ears, Ramanujan taught him the method of constructing Magic Squares, the subject of the first chapter of his Notebooks. The interest in this subject dates fromhis school days and is disconnected from the subject matter of the remainder of theNotebooks. Probably Ramanujan’s expertise in preparing the conflict free time tables for his School inspired him to a study of these Magic Squares.

Ramanujan’s investigations in continued fractions and divergent series started during this period. His betrothal to nine year old Janaki was in 1908 and his wedding took place near Karur, in 1909. Robert Kanigel , in his biography on Ramanujan, constructs a vivid account of this marriage arranged by his mother Komalattammal, not approved by his father, and dramatizes the foreboding of the impending disaster through the omens preceding the wedding, which was on the brink of being called off due to the late arrival of the bridegroom’s party.

During this period he tutored a few students in mathematics and even sought employment as a tutor in mathematics. Disappointed at the lack of recognition, during this trying period, Ramanujan had bemoaned to a friend that he was probably destined to die in poverty like Galileo! Fortunately, this was not to be.

In 1910, Ramanujan sought the patronage of Prof. V. Ramaswamy Iyer – the founder of Indian Mathematical Society – who was at Salem and asked for a clerical job in his office. The only recommendation Ramanujan had was his Notebooks which by then contained several results on Magic Squares, prime numbers, infinite series, divergent series, Bernoulli numbers, Riemann zeta function, hypergeometric series, partitions, continued fractions, elliptic functions, modular equations, etc. A scrutiny of the entries in the Notebooks was sufficient to convince Prof. Ramaswamy Iyer that Ramanujan was a gifted mathematician and he had no mind to smother his (Ramanuja’ns) genius by an appointment in the lowest rungs of the revenue department . So, he sent Ramanujan back to Madras with a letter of introduction to Prof.P.V. Seshu Aiyar, then at the Presidency College, Madras. Prof. Seshu Aiyar, who had known Ramanujan as a student at the Government Arts College, Kumbakonam, when he himself was employed there as a lecturer of mathematics, was meeting him after a gap of four years and was greatly impressed with the contents of the wellsized Notebooks. So he gave Ramanujan a note of recommendation to that true lover of mathematics, Dewan Bahadur R. Ramachandra Rao, who was then the District ollector at Nellore.

The Turning Point

With the help of a friend, R. Krishna Rao [16], who was a nephew of Dewan Bahadur Ramachandra Rao, Ramanujan went to Tirukkoilur in December 1910. This was a turning point in Ramanujan’s life. Ramachandra Rao states [17] that in the plentitude of my mathematical wisdom, I condescended to permit Ramanujan to walk into my presence. At that time, Ramanujan appeared to Ramachandra Rao as a short uncouth figure, stout, unshaved, not over-clean, with one conspicuous feature - shining eyes - walked in, with a frayed Notebook under his arm · · ·. He was miserably poor. He had run away from Kumbakonam to get leisure in Madras to pursue his studies. He never craved for any distinction. He wanted leisure, in other words, simple food to be provided for him without exertion on his part and that he should be allowed to dream on.

Though Ramachandra Rao gave him a patient hearing, he took a few days to look into the Notebooks of Ramanujan. At their fourth meeting, when Ramanujan confronted Ramachandra Rao with a letter from Prof. Saldhana of Bombay appreciating the genuineness of his work, Ramachandra Rao started to feel that Ramanujan’s work must be examined in depth by eminent mathematicians. Ramachandra Rao himself states that Ramanujan led him step-by-step to elliptic integrals and hypergeometric series and at last to his theory of divergent series not yet announced to the world and this converted him into a benefactor who undertook to underwrite Ramanujan’s expenses at Madras for some time.

Prof. Seshu Aiyar also communicated the earliest contributions of Ramanujan tothe Journal of the Indian Mathematical Society (I.M.S.) in the form of questions. These appeared in 1911 and in his brief and illustrious career Ramanujan proposed in all 59 questions or solutions to questions in this journal. The first fifteen page article entitled: Some properties of Bernoulli numbers appeared in the same 1911 volume of the journal of the I.M.S. In it Ramanujan stated eight theorems embodying arithmetical properties of the Bernoulli numbers, indicating proofs for three of them; two theorems are stated as corollaries of two others, while three theorems are stated as mere conjectures. Prof. Seshu Iyer states :

* Ramanujan’s methods were so terse and novel and his presentation was so lacking in clearness and precision, that the ordinaryreader, unaccustomed to such intellectual gymnastics, could hardly follow him.*

Ramanujan lived in a small house, called ‘Summer Hous’e, in Sami Pillai Street, Triplicane, Madras, accepting reluctantly a monthly financial assistance from the collector of Nellore for about a year. Later he declined this help and from Jan. 12 to Feb. 21, 1912, he worked as a clerk in the Accountant Generals Office, on a salary of Rs.25/- per month. Not satisfied with this job, Ramanujan applied for and secured a post in the Accounts Section (Class III, Grade IV clerk on a salray of Rs.30/- per month) in the Madras Port Trust, with the help of Mr. S. Narayana Iyer, the Manager of Port Trust, who was the treasurer of the IMS and a friend of Profs. V. Ramaswamy Aiyar and P.V. Seshu Aiyar.

Mr. Narayana Aiyer was a good mathematician and was a great source of support to Ramanujan. He was not only instrumental in Ramanujan being offered a job in the Madras Port Trust, but also in securing for Ramanujan the life-long support of Sir Francis Spring. When Ramanujan was living in No. 580, Pycrofts Road, Triplicane, Madras, he used to meet Mr. Narayana Iyer and work out Mathematics on two big slates. Narayana Aiyer’s son N. Subbanarayanan relates the role his father played in the career of Ramanujan :

My father, being a fairly good mathematician himself, was unable to capture the strides of Ramanujan’s discoveries. He used to tell him, “When I am not able to understand your steps, I do not know how other mathematicians of a critical nature will accept your genius. You must descend to my level and write at least ten steps between the two steps of yours”. Sri Ramanujan ud to say, “When it is so simple and clearto me, why should I write more steps ?” But somehow my father slowly got him round, cajoled him and made him write some more, though it used to be a mighty task of boredom to him.

Dewan Bahadur Ramachandra Rao wrote to Sir Francis Spring, Chairman of Madras Port Trust, about Ramanujan. He also induced Prof. C.L.T. Griffith of the Engineering College, Madras to take interest in Ramanujan and Prof. Griffith in turn wrote in November 1912, to Sir Francis Spring, the Chairman of Madras Port Trust about the very poor accountant who was a most remarkable mathematician and asking him to keep Ramanujan happily employed until something can be done to make use of his extraordinary gifts. As stated before, these efforts resulted in Ramanujan’s entry into Port Trust, on March 1, 1912, as a Clerk in the Accounts Department. This may well be considered as the turning point in his career prospects. He held this clerical post for 14 months. His wife joined him during this period and Ramanujan shifted his residence to Saiva Muthiah Mudali Street in George Town. This period also marked the beginning of the appreciation of his scholarship and researches in mathematics.

Prof. Griffith wrote to Prof. M.J.M. Hill, of University College, University of London, on Ramanujan’s work and he received a reply in December 1912. Unfortunately, Prof. Hill could not find time to study the results. He observed that the book which will be most useful to him is Bromwic’hs Theory of Infinite Series, published by Cambridge University Press (or Macmillan) and gave advice as to how Ramanujan could get his papers published. In a sequel to this reply, dated 7 December 1912, Prof. Hill wrote to Prof. Griffith :* Mr. Ramanujan is evidently a man with a taste for Mathematics, and with some ability, but he has got on the wrong lines. He does not understand the precautions which have to be taken in dealing with divergent series, otherwise he could not have obtained the erroneous results you send me, viz.1 + 2 + 3 + · · · + 1 = −1/12,12 + 22 + 32 + · · · +12 = 0,13 + 23 + 33 + · · · +13 = 1/240.The sums of n terms of these series are:n(n + 1)/2, n(n + 1/2)(n + 1)/3, [n(n + 1)]2/2and they all tend to 1 as n tends to 1 . I do think you can do no better for him than to get him a copy of the book I recommended, Bromwich’s Theory of Infinite Series, published by Macmillan and Co., who have branches in Calcutta and Bombay. Price 15/- net.*

It is not as though Ramanujan was not aware of the apparent absurd looking nature of the results on divergent series. Ramanujan, in his second letter to Hardy , wrote:

*I have got theorems on divergent series, theorems to calculate the convergent values corresponding to the divergent series, viz.:*

1 − 2 + 3 − 4 + · · · = 1/4,

1 − 1! + 2! − 3! + · · · = 0.596,

1 + 2 + 3 + 4 +1 = −1/12,

13 + 23 + 33 + · · · +1

3 = 1/24

.

Theorems to calculate such values for any given series (say, 1 − 11 + 22

− 33 + 44 − 55+· · ·), and the meaning of such values. I have also dealt with such questions When 9 to use, where to use, and how to use such values, where do they fail and where do they not ?

Hill failed to discern the origin of the results of Ramanujan and the three sums of the integers, their squares and their cubes are indeed the values of _(−n), for n = 1, 2, 3, respectively2.

Ramanujan published two short notes, one On question 330 of Professor Sanjana and another a Note on a set of simultaneous equations, in the IMS journal, in 1912. When Ramanujan approached Prof. Seshu Aiyar with some theorems on Prime Numbers, his attention was drawn to G.H. Hardys Tract onOrders of infinity. In it, Ramanujan observed that ([III], p.xxii): no definite expression has yet been found for the number of prime numbers less than any given number. Ramanujan told Prof. Seshu Aiyar that he ha discovered the required result. This made Prof. Seshu Aiyar suggest communication of this and other results to Mr. G.H. Hardy – a Fellow of the Royal Society and Cayley Lecturer in Mathematics at Cambridge a world famous mathematician, who was ten years Ramanujan’s senior.

The Years of Fruition

The life of Ramanujan, in the words of C.P. Snow is an admirable story, and one which showers credit on nearly everyone . Ramanujan’s first letter to Prof. Hardy, dated 16th January 1913, is a historic letter. It contained the bare statements of about 120 theorems, mostly formal identities from the Notebooks. This collection obviously represented what Ramanujan himself considered were results of importance. Ramanujan wrote:

Dear Sir,

I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of £20 per annum. I am now about 23 years of age. I have 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. I have not trodden through the conventional regular course which is followed in a University course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as ‘startling’ · · ·.

I would request you to go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the expressions that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you may give me. Requesting to be excused for the trouble I give you,

I remain, Dear Sir,

Yours truly,

(sd) S. Ramanujan.

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