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Srinivasa Ramanujan  

Born  Erode, Madras Presidency (now Tamil Nadu)  22 December 1887
Died  26 April 1920 Chetput, Madras, Madras Presidency (now Tamil Nadu)  (aged 32)
Residence  Kumbakonam, Tamil Nadu 
Nationality  Indian 
Fields  Mathematics 
Alma mater  Government Arts College Pachaiyappa's College 
Academic advisors  G. H. Hardy J. E. Littlewood 
Known for  Landau–Ramanujan constant Mock theta functions Ramanujan conjecture Ramanujan prime Ramanujan–Soldner constant Ramanujan theta function Ramanujan's sum Rogers–Ramanujan identities Ramanujan's master theorem 
Influences  G. H. Hardy 
Signature 
Srinivasa Ramanujan  

Born  Erode, Madras Presidency (now Tamil Nadu)  22 December 1887
Died  26 April 1920 Chetput, Madras, Madras Presidency (now Tamil Nadu)  (aged 32)
Residence  Kumbakonam, Tamil Nadu 
Nationality  Indian 
Fields  Mathematics 
Alma mater  Government Arts College Pachaiyappa's College 
Academic advisors  G. H. Hardy J. E. Littlewood 
Known for  Landau–Ramanujan constant Mock theta functions Ramanujan conjecture Ramanujan prime Ramanujan–Soldner constant Ramanujan theta function Ramanujan's sum Rogers–Ramanujan identities Ramanujan's master theorem 
Influences  G. H. Hardy 
Signature 
Srinivasa Ramanujan (Tamil: ஸ்ரீனிவாச ராமானுஜன்) FRS ( pronunciation (help·info)) (22 December 1887 – 26 April 1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. Living in India with no access to the larger mathematical community, which was centred in Europe at the time, Ramanujan developed his own mathematical research in isolation. As a result, he rediscovered known theorems in addition to producing new work. Ramanujan was said to be a natural genius by the English mathematician G. H. Hardy, in the same league as mathematicians such as Euler and Gauss.^{[1]}
Ramanujan was born at Erode, Madras Presidency (now Tamil Nadu) in a Tamil family.^{[2]}^{[3]}^{[unreliable source?]} His introduction to formal mathematics began at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S. L. Loney that he mastered by the age of 12; he even discovered theorems of his own, and rediscovered Euler's identity independently.^{[4]} He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan had conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant.
Ramanujan received a scholarship to study at Government College in Kumbakonam, which was later rescinded when he failed his nonmathematical coursework. He joined another college to pursue independent mathematical research. During the same time, Diwan Bahadul R. Ramachandra Rao, ICS; who was a keen mathematician and served as President of the Indian Mathematical Society himself; assisted Ramanujan in getting a clerical job in the AccountantGeneral's office at the Madras Port Trust Office to support himself.^{[5]} In 1912–1913, he sent samples of his theorems to three academics at the University of Cambridge. G. H. Hardy, recognising the brilliance of his work, invited Ramanujan to visit and work with him at Cambridge. He became a Fellow of the Royal Society and a Fellow of Trinity College, Cambridge. Ramanujan died of tuberculosis caused by illness and malnutrition in 1920 at the age of 32.
During his short life, Ramanujan independently compiled nearly 3900 results (mostly identities and equations).^{[6]} Nearly all his claims have now been proven correct, although a small number of these results were actually false and some were already known.^{[7]} 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.^{[8]} The Ramanujan Journal, an international publication, was launched to publish work in all areas of mathematics influenced by his work.^{[9]}
In December 2011, in recognition of his contribution to mathematics, the Government of India declared that Ramanujan's birth date (22 December) would be celebrated every year as National Mathematics Day and declared 2012 the National Mathematics Year.^{[10]}^{[11]}
Ramanujan was born on 22 December 1887 in Erode, Madras Presidency (now Pallipalayam, Erode, Tamil Nadu), at the residence of his maternal grandparents.^{[12]} His father, K. Srinivasa Iyengar, worked as a clerk in a sari shop and hailed from the district of Thanjavur.^{[13]} His mother, Komalatammal, was a housewife and also sang at a local temple.^{[14]} They lived in Sarangapani Street in a traditional home in the town of Kumbakonam. The family home is now a museum. When Ramanujan was a year and a half old, his mother gave birth to a son named Sadagopan, who died less than three months later. In December 1889, Ramanujan had smallpox and recovered, unlike thousands in the Thanjavur District who died from the disease that year.^{[15]} He moved with his mother to her parents' house in Kanchipuram, near Madras (now Chennai). In November 1891, and again in 1894, his mother gave birth to two children, but both children died in infancy.
On 1 October 1892, Ramanujan was enrolled at the local school.^{[16]} In March 1894, he was moved to a Telugu medium school. After his maternal grandfather lost his job as a court official in Kanchipuram,^{[17]} Ramanujan and his mother moved back to Kumbakonam and he was enrolled in the Kangayan Primary School.^{[18]} When 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 attending. His family enlisted a local constable to make sure he attended school. Within six months, Ramanujan was back in Kumbakonam.^{[18]}
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 and puranas. He learned to sing religious songs, to attend pujas at the temple and particular eating habits – all of which are part of Brahmin culture.^{[19]} At the Kangayan Primary School, Ramanujan performed well. Just before the age of 10, in November 1897, he passed his primary examinations in English, Tamil, geography and arithmetic. With his scores, he stood first in the district.^{[20]} That year, Ramanujan entered Town Higher Secondary School where he encountered formal mathematics for the first time.^{[20]}
By age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home. He was later lent a book on advanced trigonometry written by S. L. Loney.^{[4]}^{[21]} He completely mastered this book by the age of 13 and discovered sophisticated theorems on his own. By 14, he was receiving merit certificates and academic awards which continued throughout his school career and also assisted the school in the logistics of assigning its 1200 students (each with their own needs) to its 35odd teachers.^{[22]} He completed mathematical exams in half the allotted time, and showed a familiarity with geometry and infinite series. Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic.
In 1903 when he was 16, Ramanujan obtained from a friend a libraryloaned copy of a book by G. S. Carr.^{[23]}^{[24]} The book was titled A Synopsis of Elementary Results in Pure and Applied Mathematics and was a collection of 5000 theorems. Ramanujan reportedly studied the contents of the book in detail.^{[25]} The book is generally acknowledged as a key element in awakening the genius of Ramanujan.^{[25]} The next year, he had independently developed and investigated the Bernoulli numbers and had calculated the Euler–Mascheroni constant up to 15 decimal places.^{[26]} His peers at the time commented that they "rarely understood him" and "stood in respectful awe" of him.^{[22]}
When he graduated from Town Higher Secondary School 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.^{[22]} He received a scholarship to study at Government Arts College, Kumbakonam,^{[27]}^{[28]} 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.^{[29]} In August 1905, he ran away from home, heading towards Visakhapatnam and stayed in Rajahmundry^{[30]} for about a month.^{[31]} 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 Fellow of Arts 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 on the brink of starvation.^{[32]}
On 14 July 1909, Ramanujan was married to a tenyear old bride, Janakiammal (21 March 1899 – 13 April 1994).^{[33]} She came from Rajendram, a village close to Marudur (Karur district) Railway Station. Ramanujan's father did not participate in the marriage ceremony.^{[34]}
After the marriage, Ramanujan developed a hydrocele testis, an abnormal swelling of the tunica vaginalis, an internal membrane in the testicle.^{[35]} 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.^{[36]}
After his successful surgery, Ramanujan searched for a job. He stayed at friends' houses while he went 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.^{[37]}
In late 1910, Ramanujan was sick again, possibly as a result of the surgery earlier in the year. He feared for his health, and even told his friend, R. Radakrishna Iyer, to "hand these [Ramanujan's mathematical notebooks] over to Professor Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British professor Edward B. Ross, of the Madras Christian College."^{[38]} 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.^{[39]}^{[40]}
Ramanujan met deputy collector V. Ramaswamy Aiyer, who had recently founded the Indian Mathematical Society.^{[41]} Ramanujan, wishing for a job at the revenue department where Ramaswamy Aiyer worked, showed him his mathematics notebooks. As Ramaswamy Aiyer 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.^{[42]}
Ramaswamy Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras.^{[41]} 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.^{[43]}^{[44]}^{[45]} Ramachandra Rao was impressed by Ramanujan's research but doubted that it was actually his own work. Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable Bombay mathematician, in which Saldhana expressed a lack of understanding of his work but concluded that he was not a phoney.^{[46]} Ramanujan's friend, C. V. Rajagopalachari, persisted with Ramachandra Rao and tried to quell any doubts over Ramanujan's academic integrity. 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" him to a belief in Ramanujan's mathematical brilliance.^{[46]} When Rao asked him what he wanted, 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 taking care of his daily needs. Ramanujan, with the help of Ramaswamy Aiyer, had his work published in the Journal of the Indian Mathematical Society.^{[47]}
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.^{[48]} 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 his 17page paper, "Some Properties of Bernoulli's Numbers", Ramanujan gave three proofs, two corollaries and three conjectures.^{[49]} 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 lacking in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him.^{[50]}
Ramanujan later wrote another paper and also continued to provide problems in the Journal.^{[51]} In early 1912, he got a temporary job in the Madras Accountant General's office, with a salary of 20 rupees per month. He lasted for only a few weeks.^{[52]} Toward the end of that assignment he applied for a position under the Chief Accountant of the Madras Port Trust. In a letter dated 9 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.^{[53]}
Attached to his application was a recommendation from E. W. Middlemast, a mathematics professor at the Presidency College, who wrote that Ramanujan was "a young man of quite exceptional capacity in Mathematics".^{[54]} Three weeks after he had applied, on 1 March, Ramanujan learned that he had been accepted as a Class III, Grade IV accounting clerk, making 30 rupees per month.^{[55]} At his office, Ramanujan easily and quickly completed the work he was given, so he spent his spare time doing 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.
In the spring of 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to present Ramanujan's work to British mathematicians. One mathematician, M. J. M. Hill of University College London, commented that Ramanujan's papers were riddled with holes.^{[56]} He said that although Ramanujan had "a taste for mathematics, and some ability", he lacked the educational background and foundation needed to be accepted by mathematicians.^{[57]} Although Hill did not offer to take Ramanujan on 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.^{[58]}
The first two professors, H. F. Baker and E. W. Hobson, returned Ramanujan's papers without comment.^{[59]} On 16 January 1913, Ramanujan wrote to G. H. Hardy. Coming from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan's manuscripts as a possible "fraud".^{[60]} Hardy recognised some of Ramanujan's formulae but others "seemed scarcely possible to believe".^{[61]} One of the theorems Hardy found scarcely possible to believe was found on the bottom of page three (valid for 0 < a < b + 1/2):
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, and 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".^{[62]} After he saw Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy commented that "they [theorems] defeated me completely; I had never seen anything in the least like them before".^{[63]} He figured that Ramanujan's theorems "must be true, because, if they were not true, no one would have the imagination to invent them".^{[63]} Hardy asked 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".^{[64]} One colleague, E. H. Neville, later commented that "not one [theorem] could have been set in the most advanced mathematical examination in the world".^{[65]}
On 8 February 1913, Hardy wrote a letter 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".^{[66]} Before his letter arrived in Madras during the third week of February, Hardy contacted the Indian Office to plan for Ramanujan's trip to Cambridge. Secretary Arthur Davies of the Advisory Committee for Indian Students met with Ramanujan to discuss the overseas trip.^{[67]} In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to a foreign land".^{[68]} Meanwhile, Ramanujan sent a letter packed with theorems to Hardy, writing, "I have found a friend in you who views my labour sympathetically."^{[69]}
To supplement Hardy's endorsement, a former mathematical lecturer at Trinity College, Cambridge, Gilbert Walker, looked at Ramanujan's work and expressed amazement, urging him to spend time at Cambridge.^{[70]} 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".^{[71]} The board agreed to grant Ramanujan a research scholarship of 75 rupees per month for the next two years at the University of Madras.^{[72]} While he was engaged as a research student, Ramanujan continued to submit papers to the Journal of the Indian Mathematical Society. In one instance, Narayana Iyer submitted some theorems of Ramanujan on summation of series to the above mathematical journal adding "The following theorem is due to S. Ramanujan, the mathematics student of Madras University". Later in November, British Professor Edward B. Ross of Madras Christian College, whom Ramanujan had met a few years before, stormed into his class one day with his eyes glowing, asking his students, "Does Ramanujan know Polish?" The reason was that in one paper, Ramanujan had anticipated the work of a Polish mathematician whose paper had just arrived by the day's mail.^{[73]} 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 generalisations that could be made to evaluate formerly unyielding integrals.^{[74]}
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.^{[75]} Neville asked Ramanujan why he would not go 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".^{[65]} Apparently, Ramanujan's mother had a vivid dream in which the family Goddess, the deity of Namagiri, commanded her "to stand no longer between her son and the fulfilment of his life's purpose".^{[65]} Ramanujan then set sail for England, leaving his wife to stay with his parents in India.
Ramanujan boarded the S.S. Nevasa on 17 March 1914, and at 10 o'clock in the morning, the ship departed from Madras.^{[76]} He arrived in London on 14 April, 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 fiveminute walk from Hardy's room.^{[77]} Hardy and Ramanujan began to take a look at Ramanujan's 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, others had already been discovered, while the rest were new breakthroughs.^{[78]} Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, "I can believe that he's at least a Jacobi",^{[79]} while Hardy said he "can compare him only with [Leonhard] Euler or Jacobi."^{[80]}
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 Bachelor of Science degree by research (this degree was later renamed PhD) in March 1916 for his work on highly composite numbers, the first part of which was published as a paper in the Proceedings 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.^{[citation needed]} On 6 December 1917, he was elected to the London Mathematical Society. He became a Fellow of the Royal Society in 1918, becoming the second Indian to do so, following Ardaseer Cursetjee in 1841, and he was one of the youngest Fellows in the history of the Royal Society. 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.^{[81]}
Plagued by health problems throughout 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, Madras Presidency in 1919 and died soon thereafter at the age of 32. His widow, S. Janaki Ammal, moved to Mumbai, but returned to Chennai (formerly Madras) in 1950, where she lived until her death in 1994.^{[34]}
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 widespread in Madras, where Ramanujan had spent time. He had two episodes of dysentery before he left India. When not properly treated, dysentery can lie dormant for years and lead to hepatic amoebiasis,^{[5]} a difficult disease to diagnose, but once diagnosed readily cured.^{[5]}
Ramanujan has been described as a person with a somewhat shy and quiet disposition, a dignified man with pleasant manners.^{[82]} He lived a rather Spartan life while at Cambridge. Ramanujan's first Indian biographers describe him as rigorously orthodox. Ramanujan credited his acumen to his family goddess, Mahalakshmi of Namakkal. He looked to her for inspiration in his work,^{[83]} and claimed to dream of blood drops that symbolised her male consort, Narasimha, after which he would receive visions of scrolls of complex mathematical content unfolding before his eyes.^{[84]} He often said, "An equation for me has no meaning, unless it represents a thought of God."^{[85]}^{[86]}
Hardy cites Ramanujan as remarking that all religions seemed equally true to him.^{[87]} Hardy further argued that Ramanujan's religiousness had been romanticised by Westerners and overstated—in reference to his belief, not practice—by Indian biographers. At the same time, he remarked on Ramanujan's strict observance of vegetarianism.
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 by G. H. Hardy that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a byproduct, 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 = −232 with class number h(d) = 2 (note that 5×7×13×58 = 26390 and that 9801=99×99; 396=4×99) and is related to the fact that
Compare to Heegner numbers, which have class number 1 and yield similar formulae. Ramanujan's series for π converges extraordinarily rapidly (exponentially) and forms the basis of some of the fastest algorithms currently used to calculate π. Truncating the sum to the first term also gives the approximation for π, which is correct to six decimal places. See also the more general Ramanujan–Sato series.
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.^{[88]}^{[89]}
His intuition also led him to derive some previously unknown identities, such as
for all , where is the gamma function. Expanding into series of powers and equating coefficients of , , and gives some deep identities for the hyperbolic secant.
In 1918, Hardy and Ramanujan studied the partition function P(n) extensively and gave a nonconvergent 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.^{[90]}
He discovered mock theta functions in the last year of his life.^{[91]} For many years these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms.
Although there are numerous statements that could have borne 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 André Weil in algebraic geometry opened up new areas of research. That Ramanujan conjecture is an assertion on the size of the taufunction, which has as generating function the discriminant modular form Δ(q), a typical cusp form in the theory of modular forms. It was finally proven 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 in 1978 for his work on Weil conjectures.^{[92]}
While still in Madras, 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 the Madras Presidency at the time. He was also quite likely to have been influenced by the style of G. S. Carr's book studied in his youth, which stated results without proofs. Finally, it is possible that Ramanujan considered his workings to be for his personal interest alone; and therefore recorded only the results.^{[93]}
The first notebook has 351 pages with 16 somewhat organised chapters and some unorganised material. The second notebook has 256 pages in 21 chapters and 100 unorganised pages, with the third notebook containing 33 unorganised 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.^{[93]} A fourth notebook with 87 unorganised pages, the socalled "lost notebook", was rediscovered in 1976 by George Andrews.^{[5]}
Notebooks 1, 2 and 3 were published as a twovolume set in 1957 by the Tata Institute of Fundamental Research (TIFR), Mumbai, India. This was a photocopy edition of the original manuscripts, in his own handwriting.
In December 2011, as part of the celebrations of the 125th anniversary of Ramanujan's birth, TIFR republished the notebooks in a coloured twovolume collector's edition. These were produced from scanned and microfilmed images of the original manuscripts by expert archivists of Roja Muthiah Research Library, Chennai.
The number 1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a visit to the hospital to see Ramanujan. In Hardy's words:^{[94]}
“  I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No", he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."  ” 
The two different ways are
Generalizations of this idea have created the notion of "taxicab numbers". Coincidentally, 1729 is also a Carmichael number.
Hardy said : "He combined a power of generalization, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without a rival in his day. 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...".^{[95]} When asked about the methods employed by Ramanujan to arrive at his solutions, Hardy said that they were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account."^{[96]} He also stated that he had "never met his equal, and can compare him only with Euler or Jacobi."^{[96]}
Quoting K. Srinivasa Rao,^{[97]} "As for his place in the world of Mathematics, we quote Bruce C. Berndt: 'Paul Erdős has passed on to us 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.'"
Professor Bruce C. Berndt of the University of Illinois, during a lecture at IIT Madras in May 2011, stated that over the last 40 years, as nearly all of Ramanujan's theorems have been proven right, there had been a greater appreciation of Ramanujan's work and brilliance. Further, he stated Ramanujan's work was now pervading many areas of modern mathematics and physics.^{[91]}^{[98]}
In his book Scientific Edge, the physicist Jayant Narlikar spoke of "Srinivasa Ramanujan, discovered by the Cambridge mathematician 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."
During his lifelong mission in educating and propagating mathematics among the school children in India, Nigeria and elsewhere, P.K. Srinivasan has continually introduced Ramanujan's mathematical works.
Ramanujan's home state of Tamil Nadu celebrates 22 December (Ramanujan's birthday) as 'State IT Day', memorialising 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 75th anniversary of Ramanujan's birth – commemorating his achievements in the field of number theory,^{[99]} and a new design was issued on 26 December 2011, by the India Post.^{[100]}^{[101]}
Since the Centennial year of Ramanujan, every year 22 Dec, is celebrated as Ramanujan Day by the Government Arts College, Kumbakonam where he had studied and later dropped out. It is celebrated by the Department of Mathematics by organising one, two, or threeday seminars by inviting eminent scholars from universities/colleges, and participants are mainly students of mathematics, research scholars, and professors from local colleges. It was planned to celebrate the 125th birthday in a grand manner by inviting the foreign eminent mathematical scholars of this century viz., G E Andrews. and Bruce C Berndt, who are very familiar with the contributions and works of Ramanujan.
Ramanujan's work and life are celebrated on 22 December at the Indian Institute of Technology (IIT), Madras in Chennai. The Department of Mathematics celebrates this day by organising a National Symposium on Mathematical Methods and Applications (NSMMA) for one day by inviting eminent Indian and foreign scholars.
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, which nominate members of the prize committee. The Shanmugha Arts, Science, Technology & Research Academy (SASTRA), based in the state of Tamil Nadu in South India, has instituted the SASTRA Ramanujan Prize of $10,000 to be given annually to a mathematician not exceeding the age of 32 for outstanding contributions in an area of mathematics influenced by Ramanujan. The age limit refers to the years Ramanujan lived, having nevertheless still achieved many accomplishments. This prize has been awarded annually since 2005, at an international conference conducted by SASTRA in Kumbakonam, Ramanujan's hometown, around Ramanujan's birthday, 22 December.
On the 125th anniversary of his birth, India declared the birthday of Ramanujan, 22 December, as 'National Mathematics Day.' The declaration was made by Dr. Manmohan Singh in Chennai on 26 December 2011.^{[102]} Dr Manmohan Singh also declared that the year 2012 would be celebrated as the National Mathematics Year.
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