Srinivasa Aaiyangar Ramanujan

Srinivasa Aaiyangar Ramanujan

Indian mathematician (1887–1920)

Ramanujan, the son of a clerk, was born into a poor Brahmin family in Erode near Madras, India. Sometime in 1903, while a student at Kumbakonam High School, he acquired a copy of G. S. Carr's Synopsis of Elementary Results in Pure Mathematics. Carr is an unusual work, normally of use as a reference work for a professional mathematician: it consists of about 6000 theorems presented without comment, explanation, or proof. Ramanujan set himself the task of demonstrating all the formulas, a task only a natural-born mathematician would contemplate, let alone pursue. Indifferent to other subjects, Ramanujan failed every exam he entered. For a time he was supported by Ramachandra Rao, a senior civil servant and secretary of the Indian Mathematical Society (IMS). In 1912 he took a clerical position with the Madras Port Trust. At the same time it was suggested that he should seek the advice of a number of British mathematicians about his work and career.
In January 1913 Ramanujan sent a letter to a number of British mathematicians containing a number of formulas. The only one to respond was the Cambridge mathematician G. H. Hardy. Hardy noted that, while some of the formulas were familiar, others “seemed scarcely possible to believe.” Some he thought he could, with difficulty, prove himself; others, he had never seen anything like before, and they defeated Hardy completely. Despite this, it was obvious to Hardy that the formulas must be true and could only come from a mathematician of the very highest class. With Hardy's backing, Ramanujan was awarded a scholarship by the University of Madras and invited to visit Cambridge.
There were, however, religious problems facing the devout Ramanujan but these were resolved when the goddess Namagiri appeared in a dream to Ramanujan's mother absolving him from his traditional obligations. By June 1913 Ramanujan was in Cambridge working with Hardy. They collaborated on five important papers. Ramanujan was elected to the Royal Society in 1918, the first Indian to be honored in this way, and was made a fellow of Trinity College, Cambridge, in 1919. By this time his health had begun to fail. He returned to India in 1919 and died soon after from TB.
Part of Ramanujan's mathematical ability came from his ability to do mental calculations extremely quickly. It is said that he was traveling in a cab with Hardy when Hardy observed that the number of the cab in front, 1729, was a dull number. “No,” replied Ramanujan, “it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways.” (1729 = 1³+12³ and 9³+10³.)

Britannica Concise Encyclopedia: Srinivasa Aaiyangar Ramanujan

(born Dec. 22, 1887, Erode, India — died April 26, 1920, Kumbakonam) Indian mathematician. Extremely poor, he was largely self-taught from age 15. In 1913 he began a correspondence with Godfrey H. Hardy (1877 – 1947) that took him to England, where he made advances, especially in the theory of numbers, the partition of numbers, and the theory of continued fractions. He published papers in English and European journals, and in 1918 he became the first Indian elected to the Royal Society of London. He died of tuberculosis at age 32, generally unknown but recognized by mathematicians as a phenomenal genius.

Columbia Encyclopedia: Ramanujan, Srinivasa (shrē'nĭvä'sə rämä'nʊjən) , 1889–1920, Indian mathematician. He was a self-taught genius in pure mathematics who made original contributions to function theory, power series, and number theory with the training gained from a single textbook. He was invited to Cambridge by G. H. Hardy, with whom he collaborated, and continued there his work in number theory. He died of tuberculosis.

Bibliography

See R. Kanigel, The Man Who Knew Infinity (1991).

From Wikipedia

"Ramanujan" redirects here. For other uses, see Ramanujan (disambiguation).

Srinivasa Ramanujan

Born
22 December 1887(1887--)
Erode, Tamil Nadu, India
26 April 1920 (aged 32)
Chetput, (Chennai), Tamil Nadu, India

India, United Kingdom

Indian
Mathematician
University of Cambridge
G. H. Hardy and J. E. Littlewood

Landau-Ramanujan constant
Ramanujan-Soldner constant
Hindu

 

Srinivasa Ramanujan Iyengar (Tamil: ஸ்ரீனிவாச ராமானுஜன்) (22 December 1887 – 26 April 1920) was an Indian mathematician and one of the greatest mathematical geniuses of the 20th century.^[1] With almost no formal training in pure mathematics, he made substantial contributions in the areas of mathematical analysis, number theory, infinite series and continued fractions.

Ramanujan, born and raised in Erode, Tamil Nadu, India, first encountered formal mathematics at age ten. He demonstrated a natural ability at mathematics, and was given books on advanced trigonometry by S. L. Loney.^[2] He mastered this book by age thirteen, and even discovered theorems of his own. He demonstrated unusual mathematical skills at school, winning accolades and awards. By the age of seventeen, Ramanujan was conducting his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant. He received a scholarship to study at Government College in Kumbakonam. He failed his non-mathematical coursework, and lost his scholarship. He then joined another college to pursue independent mathematical research. In 1909, he married a nine-year old bride, Janaki Ammal, as per his parents' wishes (Such marriages used to be common in this particular sect of Hinduism where the husband and wife will not be allowed to stay together until the wife turns 18). To make a living, he worked as a clerk in the accountant general's office at the Madras Port Trust Office.^[1] In 1912-1913, Ramanujan sent samples of his theorems to three academics at University of Cambridge. Only G. H. Hardy recognized his brilliant work, and he asked Ramanujan to study under him at Cambridge.

Ramanujan independently compiled nearly 3900 results (mostly identities and equations) during his short lifetime.^[3] Although a small number of these results were actually false and some were already known, most of his claims have now been proven to be correct.^[4] He stated results that were both original and highly unconventional, such as the Ramanujan prime and the Ramanujan theta function, and these have inspired a vast amount of further research.^[5] However, some of his major discoveries have been rather slow to enter the mathematical mainstream. Recently, Ramanujan's formulae have found applications in the field of crystallography and in string theory. The Ramanujan Journal, an international publication, was launched to publish work in all the areas of mathematics that were influenced by Ramanujan.^[6]

Life

Childhood and early life

Ramanujan's home on Sarangapani Street, Kumbakonam.

Ramanujan was born on 22 December 1887 in Erode, Tamil Nadu, India, at the place of residence of his maternal grandparents.^[7] His father, K. Srinivasa Iyengar worked as a clerk in a sari shop and hailed from the district of Thanjavur.^[8] His mother, Komalatammal was a housewife and also a singer at a local temple. They lived in Sarangapani Street in a south-Indian-style home (now a museum) in the town of Kumbakonam. When Ramanujan was a year and a half old, his mother gave birth to a son named Sadagopan. The newborn died less than three months later. In December 1889, Ramanujan had smallpox and fortunately recovered, unlike the thousands in the Thanjavur district who had succumbed to the disease that year.^[9] He moved with his mother to her parents' house in Kanchipuram, near Madras. In November 1891, and again in 1894, his mother gave birth, but both children died before their first birthdays.

On 1 October 1892, Ramanujan was enrolled at the local school.^[10] In March 1894, he was moved to a Telugu medium school. After his maternal grandfather lost his job as a court official in Kanchipuram,^[11] Ramanujan and his mother moved back to Kumbakonam and he was enrolled in the Kangayan Primary School.^[12] After his paternal grandfather died, he was sent back to his maternal grandparents, who were now living in Madras. He did not like school in Madras, and he tried to avoid going to school. His family enlisted a local constantly to make sure he would stay in school. Within six months, Ramanujan was back in Kumbakonam again.^[12]

Since Ramanujan's father was at work most of the day, his mother took care of him as a child. He had a close relationship with her. From her, he learned about tradition, the caste system and puranas. He learned to sing religious songs, to attend pujas at the temple and eating habits — all of which were necessary for Ramanujan to be a good Brahmin child.^[13] At the Kangayan Primary School, Ramanujan performed well. Just before the age of ten, in November 1897, he passed his primary examinations in English, Tamil, geography and arithmetic. With his scores, he finished first in the district.^[14] In 1898, his mother gave birth to a healthy boy named Lakshmi Narasimhan.^[9] That year, Ramanujan entered Town Higher Secondary School where he encountered formal mathematics for the first time.^[15]

By age eleven, he had exhausted the mathematical knowledge of two college students, who were lodgers at his home. He was later lent books on advanced trigonometry written by S.L. Loney.^[16][17] He completely mastered this book by the age of thirteen and he discovered sophisticated theorems on his own. By fourteen, he achieved merit certificates and academic awards throughout his school career and also assisted the school in the logistics of assigning its 1200 students (each with their own needs) to its 35-odd teachers.^[18] He completed mathematical exams in half the allotted time, and showed a familiarity with infinite series. When he was sixteen, Ramanujan came across the book, A synopsis of elementary results in pure and applied mathematics written by George S. Carr.^[19] This book was a collection of 5000 theorems, and it introduced Ramanujan to the world of mathematics. The next year, he had independently developed and investigated the Bernoulli numbers and had calculated Euler's constant up to 15 decimal places.^[20] His peers of the time commented that they "rarely understood him" and "stood in respectful awe" of him.^[18]

When he graduated from Town High in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding student who deserved scores higher than the maximum possible marks.^[18] He received a scholarship to study at Government College in Kumbakonam,^[21] known as the "Cambridge of South India."^[22] However, Ramanujan was so intent on studying mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process.^[23] In August 1905, he ran away from home, heading towards Visakhapatnam.^[24] He later enrolled at Pachaiyappa's College in Madras. He again excelled in mathematics, but performed poorly in other subjects such as physiology. Ramanujan failed his F. A. degree exam in December 1906 and again a year later. Without a degree, he left college and continued to pursue independent research in mathematics. At this point in his life, he lived in extreme poverty and was often near the point of starvation.^[25]

Adulthood in India

On 14 July 1909, Ramanujan was married to a nine-year old bride, Janaki Ammal,^[26] as per the customs of India at that time, which included the married couple separating immediately after the ceremony, only to be reunited once the bride turns 18. After the marriage, Ramanujan developed a hydrocele testis, an abnormal swelling of the tunica vaginalis, an internal membrane in the testicle.^[27] The condition could be treated with a routine surgical operation, that would release the blocked fluid in the scrotal sac. His family did not have the money for the operation, but in January 1910, a doctor volunteered to do the surgery for free.^[28] After his successful surgery, Ramanujan searched for a job. He stayed at friends' houses while he was travelling door to door around the city of Madras (now Chennai) looking for a clerical position. To make some money, he tutored some students at Presidency College who were preparing for their F. A. exam.^[29] In late 1910, Ramanujan was sick again, possibly as a result of the surgery earlier in the year. He was fearful for his health, and he even told his friend, R. Radakrishna Iyer, to "hand these [my mathematical notebooks] over to Professor Singaravelu Mudaliar [mathematics professor at Pachaiyappa's College] or to the British professor Edward B. Ross, of the Madras Christian College."^[30] After Ramanujan recovered and got back his notebooks from Iyer, he took a northbound train from Kumbakonam to Villupuram, a coastal city under French control.^[31][32]

Getting noticed by mathematicians

He met deputy collector V. Ramaswami Iyer who had recently founded the Indian Mathematical Society.^[33] Ramanujan, wishing for a job at the revenue department where Iyer worked, showed him his mathematics notebooks. As Iyer later recalled:

“
I was struck by the extraordinary mathematical results contained in it [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department."^[34]
”

Iyer sent Ramanujan, with introduction letters, to his mathematical friends in Madras.^[33] Some of these friends looked at his work and gave him letters of introduction to R. Ramachandra Rao, the district collector for Nellore and the secretary of the Indian Mathematical Society.^[35][36][37] Ramachandra Rao was impressed by Ramanujan's work, but was doubtful that it was actually his own work. Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable Bombay (now Mumbai) mathematician, in which Saldhana expressed a lack of understanding for his work, but concluded that he was not a phony.^[38] Ramanujan's friend, C. V. Rajagopalachari, persisted with Ramachandra Rao and tried to quell any doubts over Ramanujan's academic morality. Rao agreed to give him another chance, and he listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series which Rao said ultimately "converted me" to believe Ramanujan's mathematical brilliance.^[38] Rao: "ask him what he wanted", and Ramanujan replied that he needed some work and financial support. Rao consented and sent him to Madras. He continued his mathematical research with Rao's financial aid supporting his daily needs. Ramanujan, with the help of Ramaswami Iyer, had his work published in the Journal of Indian Mathematical Society.^[39]

One of the first problems he posed in the journal was:

He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end, Ramanujan supplied the solution to the problem himself. On page 105 of his first notebook, he formulated an equation that could be used to solve the infinitely nested radicals problem.

Using this equation, the answer to the question posed in the Journal was simply 3.^[40] Ramanujan wrote his first formal paper for the Journal on the properties of Bernoulli numbers. One property he discovered was that the denominators (sequence A027642 in OEIS) of the fractions of Bernoulli numbers were always divisible by six. He also devised a method of calculating B_n based on previous Bernoulli numbers. One of these methods went as follows:

It will be observed that if n is even but not equal to zero,
(i) B_n is a fraction and the numerator of in its lowest terms is a prime number,
(ii) the denominator of B_n contains each of the factors 2 and 3 once and only once,
(iii) is an integer and consequently is an odd integer.

In "Some Properties of Bernoulli's Numbers", Ramanujan gave three proofs, two corollaries and three conjectures in his 17–page paper.^[41] Ramanujan's writing initially had many flaws. As Journal editor M. T. Narayana Iyengar noted:

“
Mr. Ramanujan's methods were so terse and novel and his presentation so lack in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him.^[42]
”

Ramanujan later wrote another paper and also continued to provide problems in the Journal.^[43] In early 1912, he got a temporary job in the Madras Accountant General's office, with a 20 rupee/month salary. He kept the job for only a few weeks.^[44] Towards the end of his job at the Account General's office, he applied for a job under the Chief Account of the Madras Port Trust. In a letter dated "9th February 1912", Ramanujan wrote:

Sir,
I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject. I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me.^[45]

Attached to his application was a recommendation from E. W. Middlemast, a mathematics professor at the Presidence College who wrote that Ramanujan was "a young man of quite exceptional capacity in Mathematics."^[46] Three weeks after he had applied, on 1 March, Ramanujan learned that he was accepted for a job as a Class III, Grade IV accounting clerk, making thirty rupees per month.^[47] At his office, Ramanujan easily and quickly completed the work he was given, so he spent his spare time doing his mathematical research. Ramanujan's boss, Sir Francis Spring, and S. Narayana Iyer, a colleague who was also treasurer of the Indian Mathematical Society, encouraged Ramanujan in his mathematical pursuits.

Contacting English mathematicians

Spring, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to expose Ramanujan's work to British mathematicians. One mathematician, M. J. M. Hill of University College in London, commented that Ramanujan's papers were riddled with holes.^[48] He said that although Ramanujan had "a taste for mathematics, and some ability," he lacked the educational background and foundation needed so that his work would be accepted by higher-up mathematicians.^[49] Although Hill did not offer to take Ramanujan in as a student, he did give thorough and serious professional advice on his work. With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.^[50]

The first two professors, H. F. Baker and E. W. Hobson, returned Ramanujan's papers without any comments.^[51] On 16 January 1913, Ramanujan wrote to G. H. Hardy, who had the foresight to quickly recognize Ramanujan's mathematical skills. The nine pages of mathematical wonder seemed like it could hardly have come from an unestablished mathematician. Hardy originally viewed Ramanujan's manuscripts as a possible "fraud."^[52] Hardy knew some of Ramanujan's formulas, but others "seemed scarcely possible to believe."^[53] One of the theorems Hardy found hard to believe was found on the bottom of page three:

Hardy was also impressed by some of Ramanujan's other work relating to infinite series:

The first result had already been determined by a mathematician named Bauer. The second one was new to Hardy. It was derived from a class of functions called a hypergeometric series which had first been researched by Leonhard Euler and Carl Friedrich Gauss. Compared to Ramanujan's work on integrals, Hardy found these results "much more intriguing."^[54] After he saw Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy commented that the "[theorems] defeated me completely; I had never seen anything in the least like them before."^[55] He figured that Ramanujan's theorems "must be true, because, if they were not true, no one would have the imagination to invent them.^[55] Hardy contacted a colleague, J. E. Littlewood, to take a look at the papers. Littlewood was amazed by the mathematical genius of Ramanujan. After discussing the papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I have received" and commented that Ramanujan was "a mathematician of the highest quality, a man of altogether exceptional originality and power."^[56] One colleague, E. H. Neville, later commented that "not one [theorem] could have been set in the most advanced mathematical examination in the world."^[57]

On 8 February 1913, Hardy wrote a letter back to Ramanujan, expressing his interest for his work. Hardy also added that it was "essential that I should see proofs of some of your assertions."^[58] Before his letter arrived in Madras during the third week of February, Hardy contacted the Indian Office to set up plans for Ramanujan's trip to Cambridge. Secretary Arthur Davies of the Advisory Committee for Indian Students met with Ramanujan to discuss the overseas trip.^[59] In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to a foreign land."^[60] Meanwhile, Ramanujan sent a letter packed with theorems to Hardy, writing, "I have found a friend in you who views my labour sympathetically."^[61]

To supplement Hardy's endorsement, a former mathematical lecturer at Trinity College in Cambridge, Gilbert Walker, looked at Ramanujan's work and expressed amazement and urged him to spend time at Cambridge.^[62] As a result of Walker's endorsement, B. Hanumantha Rao, a mathematics professor at an engineering college, invited Ramanujan's colleague Narayana Iyer to a meeting of the Board of Studies in Mathematics to discuss "what we can do for S. Ramanujan."^[63] The board met and agreed to grant Ramanujan a research scholarship of 75 rupees per month for the next two years at the University of Madras.^[64] While he was engaged as a research student, Ramanujan continued to submit papers to the Journal of the Indian Mathematical Society. In one paper, Ramanujan anticipated the work of a Polish mathematician who had published his work shortly after.^[65] In his quarterly papers, Ramanujan drew up theorems to make definite integrals more easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalizations that could be made to evaluate formerly unyielding integrals.^[66]

Hardy's correspondence with Ramanujan soured after Ramanujan refused to come to England. Hardy enlisted a colleague lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan to England.^[67] Neville asked Ramanujan why he was not coming to Cambridge. Ramanujan apparently had now accepted the proposal, as Neville put it, "Ramanujan needed no converting and that his parents' opposition had been withdrawn."^[57] Apparently, Ramanujan's friends convinced his mother to accept the journey to Cambridge. Ramanujan was personally convinced by a vivid dream his mother had, in which the family goddess Namagiri commanded her "to stand no longer between her son and the fullfilment of his life's purpose."^[57]

Life in England

Ramanujan went aboard the S. S. Nevasa on 17 March 1913, and at ten o'clock in the morning, the ship departed from Madras.^[68] He arrived in London on April 14, with E. H. Neville waiting for him with a car. Four days later, Neville took him to his house on Chesterton Road in Cambridge. Ramanujan immediately began his work with Littlewood and Hardy. After six weeks, Ramanujan moved out of Neville's house and took up residence on Whewell's Court, just a five minutes walk away from Hardy's room.^[69] Hardy and Ramanujan began to take a look at Ramanujan's work in his notebooks. Hardy had already received 120 theorems from Ramanujan in the first two letters, but there were many more results and theorems to be found in the notebooks. Hardy saw that some were wrong, some were already discovered and the rest were new breakthroughs.^[70] Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, "I can believe that he's at least a [Carl Gustav Jacob] Jacobi,"^[71] while Hardy said he "can compare him only with [Leonhard] Euler or Jacobi."^[72]

Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood and published a part of his findings there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different cultures, beliefs and working styles. Hardy was an atheist and an apostle of proof and mathematical rigour, whereas, Ramanujan was a deeply religious man and relied very strongly on his intuition. While in England, Hardy tried his best to fill the gaps in Ramanujan's education without interrupting his spell of inspiration.

Ramanujan was awarded a B.A. degree by research (this degree was later renamed PhD) in March 1916 for his work on highly composite numbers which was published as a paper in the Journal of the London Mathematical Society. The paper was over 50 pages with different properties of such numbers proven. Hardy remarked that this was one of the most unusual papers seen in Mathematical Research at that time and that Ramanujan showed extraordinary ingenuity in handling it. On 6 December 1917, he was elected to the London Mathematical Society. He was the second Indian to become a Fellow of the Royal Society in 1918 and he became one of the youngest Fellows in the entire history of the Royal Society.^[73] He was elected "for his investigation in Elliptic Functions and the Theory of Numbers." On 13 October 1918, he became the first Indian to be elected a Fellow of Trinity College, Cambridge.^[74]

Illness and return to India

Plagued by health problems all through his life, living in a country far away from home, and obsessively involved with his mathematics, Ramanujan's health worsened in England, perhaps exacerbated by stress, and by the scarcity of vegetarian food during the First World War. He was diagnosed with tuberculosis and a severe vitamin deficiency and was confined to a sanatorium. Ramanujan returned to Kumbakonam, India in 1919 and died soon thereafter at the age of 32. His wife, S. Janaki Ammal, lived in Chennai (formerly Madras) until her death in 1994.^[75]

A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D. A. B. Young concluded that it was much more likely he had hepatic amoebiasis, a parasitic infection of the liver. This is supported by the fact that Ramanujan had spent time in Madras, where the disease was widespread. He had two cases of dysentery before he left India. When not properly treated, dysentery can lie dormant for years and lead to hepatic amoebiasis.^[1] It was a difficult disease to diagnose, but once diagnosed would have been readily curable.^[1]

Personality

Ramanujan has been described as a person with a somewhat shy and quiet disposition, a dignified man with pleasant manners.^[76] He lived a rather spartan life while at Cambridge and frequently cooked vegetables alone in his room.

Spiritual life

Ramanujan was a Tamil Iyengar. His first Indian biographers describe him as rigorously orthodox. Ramanujan credited his acumen to his family goddess, Namagiri, and looked to her for inspiration in his work.^[77] He often said, "An equation for me has no meaning, unless it represents a thought of God."^[78]

Mathematical achievements

In mathematics, there is a distinction between having an insight and having a proof. Ramanujan's talent suggested a plethora of formulae that could then be investigated in depth later. It is said that Ramanujan's discoveries are unusually rich and that there is often more in it than what initially meets the eye. As a by-product, new directions of research were opened up. Examples of the most interesting of these formulae include the intriguing infinite series for π, one of which is given below

This result is based on the negative fundamental discriminant d = −4×58 with class number h(d) = 2 (note that 5×7×13×58 = 26390) and is related to the fact that,

Ramanujan's series for π converges extraordinarily rapidly (exponentially) and forms the basis of some of the fastest algorithms currently used to calculate π.

One of his remarkable capabilities was the rapid solution for problems. He was sharing a room with P.C.Mahalanobis who had a problem, "Imagine that you are on a street with houses marked 1 through n. There is a house in between (x) such that the sum of the house numbers to left of it equals the sum of the house numbers to its right. If n is between 50 and 500, what are n and x." This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist. He gave a continued fraction. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. "It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind", Ramanujan replied.

His intuition also led him to derive some previously unknown identities, such as

for all θ, where Γ(z) is the gamma function. Equating coefficients of θ⁰, θ⁴, and θ⁸ gives some deep identities for the hyperbolic secant.

In 1918, G. H. Hardy and Ramanujan studied the partition function P(n) extensively and gave a very accurate non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. Hans Rademacher, in 1937, was able to refine their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae, called the circle method.^[79]

One example of his intuition is his discovery of Mock Theta functions, in the last year of his life. This was no surprise to some mathematicians as they remarked, "He has his own creativity and the collaboration with Hardy to back it up. So, his finding these is no surprise to the mathematical community." This has gained some interest recently due to proof of the exact formula for the coefficients of any Mock Theta function. It was claimed by many mathematicians to be the most significant among his discoveries.

The Ramanujan conjecture

Main article: Ramanujan-Petersson conjecture

Although there are numerous statements that could bear the name Ramanujan conjecture, there is one statement that was very influential on later work. In particular, the connection of this conjecture with conjectures of A.Weil in algebraic geometry opened up new areas of research. That Ramanujan conjecture is an assertion on the size of the tau function, which has as generating function the discriminant modular form Δ(q), a typical cusp form in the theory of modular forms. It was finally proved in 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures. The reduction step involved is complicated. Deligne won a Fields Medal for his work on Weil conjectures.^[80]

Ramanujan's notebooks

While still in India, Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper. These results were mostly written up without any derivations. This is probably the origin of the misperception that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to make the proofs of most of his results, but chose not to.

This style of working may have been for several reasons. Since paper was very expensive, Ramanujan would do most of his work and perhaps his proofs on slate, and then transfer just the results to paper. Using a slate was common for mathematics students in India at the time. He was also quite likely to have been influenced by the style of G. S. Carr's book, which stated results without proofs. Finally, it is possible that Ramanujan considered his workings to be for his personal interest alone; and therefore only recorded the results.^[81]

The first notebook has 351 pages with 16 somewhat organized chapters and some unorganized material. The second notebook has 256 pages in 21 chapters and 100 unorganized pages, with the third notebook containing 33 unorganized pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself created papers exploring material from Ramanujan's work as did G. N. Watson, B. M. Wilson, and Bruce Berndt.^[81] A fourth notebook, the so-called "lost notebook", was rediscovered in 1976 by George Andrews.^[1]

Other mathematicians' views of Ramanujan

Ramanujan is generally hailed as an all time great like Euler, Gauss or Jacobi for his natural mathematical genius⁵. G. H. Hardy quotes: "The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet 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...".^[82] Hardy went on to claim that his greatest contribution to mathematics came from Ramanujan.

Quoting K. Srinivasa Rao,^[83] "As for his place in the world of Mathematics, we quote Bruce C. Berndt: 'Paul Erdős has passed on to us G. H. Hardy's personal ratings of mathematicians. 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, J.E. Littlewood 30, David Hilbert 80 and Ramanujan 100.'"

In his book Scientific Edge, noted physicist Jayant Narlikar stated that "Srinivasa Ramanujan, discovered by the Cambridge mathematician G.H. Hardy, whose great mathematical findings were beginning to be appreciated from 1915 to 1919. His achievements were to be fully understood much later, well after his untimely death in 1920. For example, his work on the highly composite numbers (numbers with a large number of factors) started a whole new line of investigations in the theory of such numbers." Narlikar also goes on to say that his work was one of the top ten achievements of 20^th century Indian science and "could be considered in the Nobel Prize class."^[84] The work of other 20^th century Indian scientists which Narlikar considered to be of Nobel Prize class were those of Chandrasekhara Venkata Raman, Megh Nad Saha and Satyendra Nath Bose.

Recognition

Ramanujan's home state of Tamil Nadu celebrates December 22 (Ramanujan's birthday) as 'State IT Day', memorializing both the man and his achievements, as a native of Tamil Nadu. A stamp picturing Ramanujan was released by the Government of India in 1962 — the 75^th anniversary of Ramanujan's birth — commemorating his achievements in the field of number theory.

A prize for young mathematicians from developing countries has been created in the name of Ramanujan by the International Centre for Theoretical Physics (ICTP), in cooperation with the International Mathematical Union, who nominate members of the prize committee. During the year 1987 (Ramanujan's centennial), the printed form of Ramanujan's Lost Notebook by the Narosa publishing house of [[Springer Science+Business Media|Springer-Verlag]] was released by the late Indian prime minister, Rajiv Gandhi, who presented the first copy to S. Janaki Ammal Ramanujan (Ramanujan's late widow) and the second copy to George Andrews in recognition of his contributions in the field of number theory.

Projected films

• An international feature film on Ramanujan's life will begin shooting in 2007 in Tamil Nadu state and Cambridge. It is being produced by an Indo-British collaboration; it will be co-directed by Stephen Fry and Dev Benegal.^[85] A play First Class Man by Alter Ego Productions ^[86] was based on David Freeman's "First Class Man". The play is centered around Ramanujan and his complex and dysfunctional relationship with G. H. Hardy.
• Another film based on the book The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel is being made by Edward Pressman and Matthew Brown.