Srinivasa Ramanujan

"Ramanujan" redirects here. For other uses, see Ramanujan (disambiguation).
Srinivasa Ramanujan
Born 22 December 1887
Died 26 April 1920 (aged 32)
London, United Kingdom
Nationality Indian
Fields Mathematics
Institutions Trinity College, Cambridge
Alma mater Government Arts College (no degree)
Pachaiyappa's College (no degree)
Trinity College, Cambridge (BSc, 1916)
Thesis Highly Composite Numbers (1916)
J. E. Littlewood
Known for
Influences G. S. Carr
Influenced G. H. Hardy
Notable awards Fellow of the Royal Society
Signature
In this Indian name, the name Srinivasa is a patronymic, not a family name, and the person should be referred to by the given name, Ramanujan.

Srinivasa Iyengar Ramanujan FRS (pronunciation: /ˈʃrniˌvɑːsə ˈrɑːmɑːˌnʊən/; 22 December 1887 – 26 April 1920) was an Indian mathematician and autodidact who lived during the British Raj. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions. Ramanujan initially developed his own mathematical research in isolation; it was quickly recognized by Indian mathematicians. When his skills became obvious and known to the wider mathematical community, centred in Europe at the time, he began a partnership with the English mathematician G. H. Hardy. The Cambridge professor realized that Srinivasa Ramanujan had produced new theorems in addition to rediscovering previously known ones.

During his short life, Ramanujan independently compiled nearly 3,900 results (mostly identities and equations).[1] Nearly all his claims have now been proven correct.[2] His original and highly unconventional results, such as the Ramanujan prime and the Ramanujan theta function, have inspired a vast amount of further research.[3] The Ramanujan Journal, a peer-reviewed scientific journal, was established to publish work in all areas of mathematics influenced by Ramanujan.[4]

Deeply religious,[5] Ramanujan credited his substantial mathematical capacities to divinity: '"An equation for me has no meaning," he once said, "unless it expresses a thought of God."'[6]

Name

The name Ramanujan means "younger brother of the god Rama."[7] The name Srinivasa is a combination of Sri and Nivasa; Sri refers to the female energy of God while Nivasa means a living place, so the word Srinivasa literally means the place where the female energy of the God lives."[8] Iyengar is a caste of Hindu Brahmins of Tamil origin whose members follow the Visishtadvaita philosophy propounded by Ramanuja; there are several opinions concerning its etymology.

Early life

Ramanujan's home on Sarangapani Sannidhi Street, Kumbakonam

Ramanujan was born on 22 December 1887 into a Tamil Brahmin Iyengar family in Erode, Madras Presidency (now Tamil Nadu), at the residence of his maternal grandparents.[9] His father, K. Srinivasa Iyengar, worked as a clerk in a sari shop and hailed from Thanjavur district.[10] His mother, Komalatammal, was a housewife and also sang at a local temple.[11] They lived in a small traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam.[12] The family home is now a museum. When Ramanujan was a year and a half old, his mother gave birth to a son, Sadagopan, who died less than three months later. In December 1889, Ramanujan contracted smallpox, but unlike the thousands in the Thanjavur district who died of the disease that year, he recovered.[13] He moved with his mother to her parents' house in Kanchipuram, near Madras (now Chennai). His mother gave birth to two more children, in 1891 and 1894, but both died in infancy.

On 1 October 1892, Ramanujan was enrolled at the local school.[14] After his maternal grandfather lost his job as a court official in Kanchipuram,[15] Ramanujan and his mother moved back to Kumbakonam and he was enrolled in the Kangayan Primary School.[16] When his paternal grandfather died, he was sent back to his maternal grandparents, then living in Madras. He did not like school in Madras, and tried to avoid attending. His family enlisted a local constable to make sure the boy attended school. Within six months, Ramanujan was back in Kumbakonam.[16]

Since Ramanujan's father was at work most of the day, his mother took care of the boy 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 to maintain particular eating habits – all of which are part of Brahmin culture.[17] At the Kangayan Primary School, Ramanujan performed well. Just before turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography and arithmetic with the best scores in the district.[18] That year, Ramanujan entered Town Higher Secondary School, where he encountered formal mathematics for the first time.[18]

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 by S. L. Loney on advanced trigonometry.[19][20] He mastered this by the age of 13 while discovering sophisticated theorems on his own. By 14, he was receiving merit certificates and academic awards that continued throughout his school career, and he assisted the school in the logistics of assigning its 1200 students (each with differing needs) to its 35-odd teachers.[21] 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; he developed his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried to do so.

In 1903, when he was 16, Ramanujan obtained from a friend a library copy of a A Synopsis of Elementary Results in Pure and Applied Mathematics, G. S. Carr's collection of 5,000 theorems.[22][23] Ramanujan reportedly studied the contents of the book in detail.[24] The book is generally acknowledged as a key element in awakening his genius.[24] The next year, Ramanujan independently developed and investigated the Bernoulli numbers and calculated the Euler–Mascheroni constant up to 15 decimal places.[25] His peers at the time commented that they "rarely understood him" and "stood in respectful awe" of him.[21]

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.[21] He received a scholarship to study at Government Arts College, Kumbakonam,[26][27] but was so intent on mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship in the process.[28] In August 1905, Ramanujan ran away from home, heading towards Visakhapatnam, and stayed in Rajahmundry[29] for about a month.[30] He later enrolled at Pachaiyappa's College in Madras. There he passed in mathematics, choosing only to attempt questions that appealed to him and leaving the rest unanswered, but performed poorly in other subjects, such as English, physiology and Sanskrit.[31] Ramanujan failed his Fellow of Arts exam in December 1906 and again a year later. Without a FA degree, he left college and continued to pursue independent research in mathematics, living in extreme poverty and often on the brink of starvation.[32]

It was in 1910, after a meeting between the 23-year-old Ramanujan and the founder of the Indian Mathematical Society, V. Ramaswamy Aiyer, also known as Professor Ramaswami, that Ramanujan started to get recognition within the mathematics circles of Madras, subsequently leading to his inclusion as a researcher at the University of Madras.[33]

On 14 July 1909, Ramanujan married Janaki (Janakiammal) (21 March 1899 – 13 April 1994), then a ten-year-old girl whom his mother had selected for him a year earlier.[34][35] It was not unusual for marriages to be arranged with young girls. Some sources claim Janaki was ten years old when they married.[36] She came from Rajendram, a village close to Marudur (Karur district) Railway Station. Ramanujan's father did not participate in the marriage ceremony.[37] As was common at that time, Janakiammal continued to stay at her maternal home for three years after marriage till she attained puberty. In 1912, when Janaki was twelve years old, she and Ramanujan's mother joined Ramanujan in Madras.[38]

After the marriage, Ramanujan developed a hydrocele testis.[39] The condition could be treated with a routine surgical operation that would release the blocked fluid in the scrotal sac, but his family did not have the money for the operation. In January 1910, a doctor volunteered to do the surgery for free.[40]

After his successful surgery, Ramanujan searched for a job. He stayed at a friends' house while he went from door to door around Madras looking for a clerical position. To make money, he tutored students at Presidency College who were preparing for their F.A. exam.[41]

In late 1910, Ramanujan was sick again. He feared for his health, and 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."[42] After Ramanujan recovered and retrieved his notebooks from Iyer, he took a train from Kumbakonam to Villupuram, a coastal city under French control.[43][44] In 1912, Ramanujan moved to a house in Saiva Muthaiah Mudali street, George Town, Madras with his wife and mother where they lived for a few months.[45] In May 1913, upon securing a research position at Madras University, Ramanujan moved with his family to Triplicane.[46]

Attention towards mathematics

Ramanujan met deputy collector V. Ramaswamy Aiyer, who had founded the Indian Mathematical Society.[47] Wishing for a job at the revenue department where Aiyer worked, Ramanujan showed him his mathematics notebooks. As 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.[48]

Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras.[47] Some of them 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.[49][50][51] Rao was impressed by Ramanujan's research but doubted that it was 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 phony.[52] Ramanujan's friend C. V. Rajagopalachari tried to quell Rao's doubts about Ramanujan's academic integrity. Rao agreed to give him another chance, and 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 brilliance.[52] When Rao asked him what he wanted, Ramanujan replied that he needed work and financial support. Rao consented and sent him to Madras. He continued his research, with Rao's financial aid taking care of his daily needs. With Aiyer's help, Ramanujan had his work published in the Journal of the Indian Mathematical Society.[53]

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

${\displaystyle {\sqrt {1+2{\sqrt {1+3{\sqrt {1+\cdots }}}}}}.}$

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.

${\displaystyle x+n+a={\sqrt {ax+(n+a)^{2}+x{\sqrt {a(x+n)+(n+a)^{2}+(x+n){\sqrt {\cdots }}}}}}}$

Using this equation, the answer to the question posed in the Journal was simply 3, obtained by setting x = 2, n = 1, and a = 0.[54] 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 the OEIS) of the fractions of Bernoulli numbers were always divisible by six. He also devised a method of calculating Bn based on previous Bernoulli numbers. One of these methods follows:

It will be observed that if n is even but not equal to zero,

1. Bn is a fraction and the numerator of Bn/n in its lowest terms is a prime number,
2. the denominator of Bn contains each of the factors 2 and 3 once and only once,
3. 2n(2n − 1)Bn/n is an integer and 2(2n − 1)Bn consequently is an odd integer.

In his 17-page paper, "Some Properties of Bernoulli's Numbers" (1911), Ramanujan gave three proofs, two corollaries and three conjectures.[55] 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.[56]

Ramanujan later wrote another paper and also continued to provide problems in the Journal.[57] 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 only a few weeks.[58] 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.[59]

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".[60] 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.[61] 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.

Contacting British mathematicians

In the spring of 1913, Narayana Iyer, Ramachandra Rao and E. W. Middlemast tried to present Ramanujan's work to British mathematicians. M. J. M. Hill of University College London commented that Ramanujan's papers were riddled with holes.[62] 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.[63] 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.[64]

The first two professors, H. F. Baker and E. W. Hobson, returned Ramanujan's papers without comment.[65] 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.[66] Hardy recognised some of Ramanujan's formulae but others "seemed scarcely possible to believe".[67] One of the theorems Hardy found amazing was on the bottom of page three (valid for 0 < a < b + 1/2):

${\displaystyle \int \limits _{0}^{\infty }{\frac {1+{\dfrac {x^{2}}{(b+1)^{2}}}}{1+{\dfrac {x^{2}}{a^{2}}}}}\times {\frac {1+{\dfrac {x^{2}}{(b+2)^{2}}}}{1+{\dfrac {x^{2}}{(a+1)^{2}}}}}\times \cdots \,dx={\frac {\sqrt {\pi }}{2}}\times {\frac {\Gamma \left(a+{\frac {1}{2}}\right)\Gamma (b+1)\Gamma (b-a+1)}{\Gamma (a)\Gamma \left(b+{\frac {1}{2}}\right)\Gamma \left(b-a+{\frac {1}{2}}\right)}}.}$

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

${\displaystyle 1-5\left({\frac {1}{2}}\right)^{3}+9\left({\frac {1\times 3}{2\times 4}}\right)^{3}-13\left({\frac {1\times 3\times 5}{2\times 4\times 6}}\right)^{3}+\cdots ={\frac {2}{\pi }}}$
${\displaystyle 1+9\left({\frac {1}{4}}\right)^{4}+17\left({\frac {1\times 5}{4\times 8}}\right)^{4}+25\left({\frac {1\times 5\times 9}{4\times 8\times 12}}\right)^{4}+\cdots ={\frac {2{\sqrt {2}}}{{\sqrt {\pi }}\,\Gamma ^{2}\left({\frac {3}{4}}\right)}}.}$

The first result had already been determined by G. Bauer in 1859. The second was new to Hardy, and was derived from a class of functions called hypergeometric series, which had first been researched by Leonhard Euler and Carl Friedrich Gauss. Hardy found these results "much more intriguing" than Gauss's work on integrals.[68] After seeing 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".[69] He figured that Ramanujan's theorems "must be true, because, if they were not true, no one would have the imagination to invent them".[69] Hardy asked a colleague, J. E. Littlewood, to take a look at the papers. Littlewood was amazed by Ramanujan's genius. After discussing the papers with Littlewood, Hardy concluded that the letters were "certainly the most remarkable I have received" and said that Ramanujan was "a mathematician of the highest quality, a man of altogether exceptional originality and power".[70] One colleague, E. H. Neville, later remarked that "not one [theorem] could have been set in the most advanced mathematical examination in the world".[71]

On 8 February 1913, Hardy wrote Ramanujan a letter expressing his interest in his work, adding that it was "essential that I should see proofs of some of your assertions".[72] 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.[73] In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to a foreign land".[74] Meanwhile, he sent Hardy a letter packed with theorems, writing, "I have found a friend in you who views my labour sympathetically."[75]

To supplement Hardy's endorsement, Gilbert Walker, a former mathematical lecturer at Trinity College, Cambridge, looked at Ramanujan's work and expressed amazement, urging the young man to spend time at Cambridge.[76] 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".[77] The board agreed to grant Ramanujan a research scholarship of 75 rupees per month for the next two years at the University of Madras.[78] 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 of Ramanujan's theorems on summation of series to the 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 in the day's mail.[79] 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.[80]

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.[81] 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".[71] 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".[71] Ramanujan voyaged to England by ship, leaving his wife to stay with his parents in India.

Life in England

Ramanujan (centre) with other scientists at Trinity College
Whewell's Court, Trinity College, Cambridge

Ramanujan departed from Madras aboard the S.S. Nevasa on 17 March 1914.[82] When he disembarked in London on 14 April, Neville was 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, a five-minute walk from Hardy's room.[83] Hardy and Littlewood began to 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 in the notebooks. Hardy saw that some were wrong, others had already been discovered, and the rest were new breakthroughs.[84] Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, "I can believe that he's at least a Jacobi",[85] while Hardy said he "can compare him only with Euler or Jacobi."[86]

Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood, and published 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 who relied very strongly on his intuition. While in England, Hardy tried his best to fill the gaps in Ramanujan's education without interrupting his 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 more than 50 pages and proved various properties of such numbers. Hardy remarked that it 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. In 1918 he was elected a Fellow of the Royal Society, the second Indian admitted to the Royal Society, following Ardaseer Cursetjee in 1841. At age 31 Ramanujan 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 was the first Indian to be elected a Fellow of Trinity College, Cambridge.[87]

Illness and death

Throughout his life, Ramanujan was plagued by health problems. His health worsened in England. He was diagnosed with tuberculosis and a severe vitamin deficiency, and was confined to a sanatorium. In 1919 he returned to Kumbakonam, Madras Presidency, and soon thereafter, in 1920, died at the age of 32. After his death, his brother Tirunarayanan chronicled Ramanujan's remaining handwritten notes consisting of formulae on singular moduli, hypergeometric series and continued fractions and compiled them.[38] Ramanujan's widow, Smt. Janaki Ammal, moved to Bombay; in 1950 she returned to Madras, where she lived in Triplicane until her death in 1994 at the age 95.[37][38]

A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D. A. B. Young[88] concluded that it was much more likely he had hepatic amoebiasis, an illness then widespread in Madras, rather than tuberculosis. 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.[89] Amoebiasis was a treatable and often curable disease at the time.[89][90]

Personality and spiritual life

Ramanujan has been described as a person of a somewhat shy and quiet disposition, a dignified man with pleasant manners.[91] He lived a rather spartan life at Cambridge. Ramanujan's first Indian biographers describe him as a rigorously orthodox Hindu. He credited his acumen to his family goddess, Mahalakshmi of Namakkal. He looked to her for inspiration in his work[92] and said he dreamed of blood drops that symbolised her male consort, Narasimha. Afterward he would receive visions of scrolls of complex mathematical content unfolding before his eyes.[93] He often said, "An equation for me has no meaning unless it represents a thought of God."[94]

Hardy cites Ramanujan as remarking that all religions seemed equally true to him.[95] Hardy further argued that Ramanujan's religious belief 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 vegetarianism.[96]

Mathematical achievements

In mathematics, there is a distinction between having an insight and having a proof. Ramanujan proposed an abundance of formulae that could be investigated later in depth. G. H. Hardy said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a byproduct of his work, 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:

${\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}.}$

This result is based on the negative fundamental discriminant d = −4 × 58 = −232 with class number h(d) = 2. 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 3962 and is related to the fact that

${\textstyle e^{\pi {\sqrt {58}}}=396^{4}-104.000000177\dots .}$

This might be compared 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 98012/4412 for π, which is correct to six decimal places. See also the more general Ramanujan–Sato series.

One of Ramanujan's remarkable capabilities was the rapid solution of problems. Once, a roommate of his, P. C. Mahalanobis, posed the following 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 the 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.[97][98]

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

${\displaystyle \left(1+2\sum _{n=1}^{\infty }{\frac {\cos(n\theta )}{\cosh(n\pi )}}\right)^{-2}+\left(1+2\sum _{n=1}^{\infty }{\frac {\cosh(n\theta )}{\cosh(n\pi )}}\right)^{-2}={\frac {2\Gamma ^{4}\left({\frac {3}{4}}\right)}{\pi }}={\frac {8\pi ^{3}}{\Gamma ^{4}\left({\frac {1}{4}}\right)}}}$

for all θ, where Γ(z) is the gamma function, and related to a special value of the Dedekind eta function. Expanding into series of powers and equating coefficients of θ0, θ4, and θ8 gives some deep identities for the hyperbolic secant.

In 1918 Hardy and Ramanujan studied the partition function P(n) extensively. They gave a 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.[99]

In the last year of his life, Ramanujan discovered mock theta functions.[100] For many years these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms.

The Ramanujan conjecture

Although there are numerous statements that could have borne the name Ramanujan conjecture, there is one that was highly 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 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 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 that work.[101]

In his paper "On certain arithmetical functions", Ramanujan defined the so-called delta-function whose coefficients are called τ(n) (the Ramanujan tau function).[102] He proved many congruences for these numbers such as τ(p) ≡ 1 + p11 mod 691 for primes p. This congruence (and others like it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) to conjecture that there is a theory of Galois representations which "explains" these congruences and more generally all modular forms. Δ(z) is the first example of a modular form to be studied in this way. Pierre Deligne (in his Fields Medal winning work) proved Serre's conjecture. The proof of Fermat's Last Theorem proceeds by first reinterpreting elliptic curves and modular forms in terms of these Galois representations. Without this theory there would be no proof of Fermat's Last Theorem.[103]

Ramanujan's notebooks

Further information: Ramanujan's lost notebook

While still in Madras, Ramanujan recorded the bulk of his results in four notebooks of loose-leaf paper. They 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 prove most of his results, but chose not to.

That 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, 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.[104]

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.[104] A fourth notebook with 87 unorganised pages, the so-called "lost notebook", was rediscovered in 1976 by George Andrews.[89]

Notebooks 1, 2 and 3 were published as a two-volume 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 two-volume collector's edition. These were produced from scanned and microfilmed images of the original manuscripts by expert archivists of Raja Muthiah Research Library, Chennai.

Hardy–Ramanujan number 1729

Main article: 1729 (number)

The number 1729 is known as the Hardy–Ramanujan number after a famous visit by Hardy to see Ramanujan at a hospital. In Hardy's words:[105]

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."

Immediately before this anecdote, Hardy quoted Littlewood as saying, "Every positive integer was one of [Ramanujan's] personal friends."[106]

The two different ways are

1729 = 13 + 123 = 93 + 103.

Generalizations of this idea have created the notion of "taxicab numbers".

Other mathematicians' views of Ramanujan

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...".[107] When asked about the methods Ramanujan employed 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."[108] He also stated that he had "never met his equal, and can compare him only with Euler or Jacobi."[108]

K. Srinivasa Rao has said,[109] "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.'" During a lecture at IIT Madras in May 2011, Berndt stated that over the last 40 years, as nearly all of Ramanujan's theorems have been proven right, there had been greater appreciation of Ramanujan's work and brilliance, and that Ramanujan's work was now pervading many areas of modern mathematics and physics.[100][110]

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."

Posthumous recognition

Bust of Ramanujan in the garden of Birla Industrial & Technological Museum

Ramanujan's home state of Tamil Nadu celebrates 22 December (Ramanujan's birthday) as 'State IT Day'. 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,[111] and a new design was issued on 26 December 2011, by the India Post.[112][113]

Since Ramanujan's centennial year, his birthday, 22 December, has been annually celebrated as Ramanujan Day by the Government Arts College, Kumbakonam where he studied and at the IIT Madras in Chennai. A prize for young mathematicians from developing countries has been created in Ramanujan's name by the International Centre for Theoretical Physics (ICTP) in cooperation with the International Mathematical Union, which nominate members of the prize committee. The SASTRA University, based in the state of Tamil Nadu in South India, has instituted the SASTRA Ramanujan Prize of US\$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. Vasavi College of Engineering named its Department of Computer Science and Information Technology "Ramanujan Block".

In 2011, on the 125th anniversary of his birth, the Indian Government declared that 22 December will be celebrated every year as National Mathematics Day.[114] Then Indian Prime Minister Manmohan Singh also declared that the year 2012 would be celebrated as the National Mathematics Year.

Selected publications on works of Ramanujan

• Ramanujan, Srinivasa; Hardy, G. H.; Seshu Aiyar, P. V.; Wilson, B. M.; Berndt, Bruce C. (2000). Collected Papers of Srinivasa Ramanujan. AMS. ISBN 0-8218-2076-1.
This book was originally published in 1927 after Ramanujan's death. It contains the 37 papers published in professional journals by Ramanujan during his lifetime. The third reprint contains additional commentary by Bruce C. Berndt.
• S. Ramanujan (1957). Notebooks (2 Volumes). Bombay: Tata Institute of Fundamental Research.
These books contain photocopies of the original notebooks as written by Ramanujan.
• S. Ramanujan (1988). The Lost Notebook and Other Unpublished Papers. New Delhi: Narosa. ISBN 3-540-18726-X.
This book contains photo copies of the pages of the "Lost Notebook".
• Problems posed by Ramanujan, Journal of the Indian Mathematical Society.
• S. Ramanujan (2012). Notebooks (2 Volumes). Bombay: Tata Institute of Fundamental Research.
This was produced from scanned and microfilmed images of the original manuscripts by expert archivists of Roja Muthiah Research Library, Chennai.

Notes

1. ^ Berndt, Bruce C. (1997-12-12). Ramanujan's Notebooks, Part 5. Springer Science & Business Media. p. 4. ISBN 9780387949413.
2. ^ "Rediscovering Ramanujan". Frontline. 16 (17): 650. August 1999. Retrieved 20 December 2012.
3. ^ Ono, Ken (June–July 2006). "Honoring a Gift from Kumbakonam" (PDF). Notices of the American Mathematical Society. Mathematical Association of America. 53 (6): 650. Retrieved 23 June 2007.
4. ^ Alladi, Krishnaswami; Elliott, P. D. T. A.; Granville, A. (30 September 1998). Analytic and Elementary Number Theory: A Tribute to Mathematical Legend Paul Erdos. Springer Science & Business Media. p. 6. ISBN 9780792382737.
5. ^ Kanigel, Robert (2016-04-26). The Man Who Knew Infinity: A Life of the Genius Ramanujan. Simon and Schuster. pp. 30–33. ISBN 9781476763491.
6. ^ "The Man Who Knew Infinity," (1991), Kanigel, Robert, page 7 of Prologue
7. ^ "The Man Who Knew Infinity," 1991, Kanigel, Robert (page 12)
8. ^ http://www.sjsu.edu/faculty/watkins/ramanujan.htm
9. ^ Kanigel 1991, p. 11
10. ^ Kanigel 1991, pp. 17–18
11. ^ Berndt & Rankin 2001, p. 89
12. ^ Srinivasan, Pankaja (19 October 2012). "The Nostalgia Formula". The Hindu. Retrieved 7 September 2016.
13. ^ Kanigel 1991, p. 12
14. ^ Kanigel 1991, p. 13
15. ^ Kanigel 1991, p. 19
16. ^ a b Kanigel 1991, p. 14
17. ^ Kanigel 1991, p. 20
18. ^ a b Kanigel 1991, p. 25
19. ^ Berndt & Rankin 2001, p. 9
20. ^ Hardy, G. H. (1999). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Providence, Rhode Island: American Mathematical Society. p. 2. ISBN 0-8218-2023-0.
21. ^ a b c Kanigel 1991, p. 27
22. ^ Kanigel 1991, p. 39
23. ^ A to Z of mathematicians by Tucker McElroy 2005 ISBN 0-8160-5338-3-page 221
24. ^ a b Ramanujan Aiyangar, Srinivasa; Hardy, Godfrey Harold; Aiyar, P. Veṅkatesvara Seshu (2000), "Collected papers of Srinivasa Ramanujan", Nature, 123 (3104): xii, Bibcode:1929Natur.123..631L, doi:10.1038/123631a0, ISBN 0-8218-2076-1
25. ^ Kanigel 1991, p. 90
26. ^ Kanigel 1991, p. 28
27. ^ Kanigel 1991, p. 45
28. ^ Kanigel 1991, p. 47
29. ^ "Ramanujan lost and found: a 1905 letter from The Hindu". The Hindu. Chennai, India. 25 December 2011.
30. ^ Kanigel 1991, pp. 48–49
31. ^ Krishnamachari, Suganthi (27 June 2013). "Travails of a Genius". The Hindu. Retrieved 7 September 2016.
32. ^ Kanigel 1991, pp. 55–56
33. ^ Krishnamurthy, Prof. V. "Srinivasa Ramanujan - His life and his genius". www.krishnamurthys.com. (Expository address delivered on Sep.16, 1987 at Visvesvarayya Auditorium as part of the celebrations of Ramanujan Centenary by the IISC, Bangalore). Retrieved 7 September 2016.
34. ^ Kolata 1987, p. 236
35. ^ Kanigel 1991, p. 71
36. ^ Bullough, V.L. (1990). History in adult human sexual behavior with children and adolescents in Western societies (Pedophilia: Biosocial Dimensions ed.). Springer-Verlag New York Inc. p. 71. ISBN 9781461396840. Retrieved 27 April 2016.
37. ^ a b "Ramanujan's wife: Janakiammal (Janaki)" (PDF). Institute of Mathematical Sciences, Chennai. Retrieved 10 November 2012.
38. ^ a b c Janardhanan, Arun (6 December 2015). "A passage to infinity". Indian Express. Retrieved 7 September 2016.
39. ^ Kanigel 1991, p. 72
40. ^ Ramanujan, Srinivasa (1968). P. K. Srinivasan, ed. Ramanujan Memorial Number: Letters and Reminiscences. Madras: Muthialpet High School. Vol. 1, p100.
41. ^ Kanigel 1991, p. 73
42. ^ Kanigel 1991, pp. 74–75
43. ^ Ranganathan, Shiyali Ramamrita (1967). Ramanujan: The Man and the Mathematician. Bombay: Asia Publishing House., p. 23.
44. ^ Srinivasan (1968), Vol. 1, p99.
45. ^ Rao. "Ramanujan's wife Janakiammal (Janaki)" (PDF). http://www.imsc.res.in. IMSC (Institute of mathematical Sciences), Chennai. Retrieved 7 September 2016. External link in |website= (help)
46. ^ "About Ramanujan". http://ramanujaninstitute.com. The Ramanujan Institute. Retrieved 7 September 2016. External link in |website= (help)
47. ^ a b Kanigel 1991, p. 77
48. ^ Srinivasan (1968), Vol. 1, p129.
49. ^ Srinivasan (1968), Vol. 1, p86.
50. ^ Neville, Eric Harold (January 1921). "The Late Srinivasa Ramanujan". Nature. 106 (2673): 661–662. Bibcode:1921Natur.106..661N. doi:10.1038/106661b0.
51. ^ Ranganathan 1967, p. 24
52. ^ a b Kanigel 1991, p. 80
53. ^ Kanigel 1991, p. 86
54. ^ Kanigel 1991, p. 87
55. ^ Kanigel 1991, p. 91
56. ^ Seshu Iyer, P. V. (June 1920). "The Late Mr. S. Ramanujan, B.A., F.R.S.". Journal of the Indian Mathematical Society. 12 (3): 83.
57. ^ Neville (March 1942), p292.
58. ^ Srinivasan (1968), p176.
59. ^ Srinivasan (1968), p31.
60. ^ Srinivasan (1968), p49.
61. ^ Kanigel 1991, p. 96
62. ^ Kanigel 1991, p. 105
63. ^ Letter from M. J. M. Hill to a C. L. T. Griffith (a former student who sent the request to Hill on Ramanujan's behalf), 28 November 1912.
64. ^ Kanigel 1991, p. 106
65. ^ Kanigel 1991, pp. 170–171
66. ^ Snow, C. P. (1966). Variety of Men. New York: Charles Scribner's Sons. pp. 30–31.
67. ^ Hardy, G. H. (June 1920). "Obituary, S. Ramanujan". Nature. 105 (7): 494. Bibcode:1920Natur.105..494H. doi:10.1038/105494a0.
68. ^ Kanigel 1991, p. 167
69. ^ a b Kanigel 1991, p. 168
70. ^ Hardy (June 1920), pp494–495.
71. ^ a b c Neville, Eric Harold (March 1942). "Srinivasa Ramanujan". Nature. 149 (3776): 293. Bibcode:1942Natur.149..292N. doi:10.1038/149292a0.
72. ^ Letter, Hardy to Ramanujan, 8 February 1913.
73. ^ Letter, Ramanujan to Hardy, 22 January 1914.
74. ^ Kanigel 1991, p. 185
75. ^ Letter, Ramanujan to Hardy, 27 February 1913, Cambridge University Library.
76. ^ Kanigel 1991, p. 175
77. ^ Ram, Suresh (1972). Srinivasa Ramanujan. New Delhi: National Book Trust. p. 29.
78. ^ Ranganathan 1967, pp. 30–31
79. ^ Ranganathan 1967, p. 12
80. ^ Kanigel 1991, p. 183
81. ^ Kanigel 1991, p. 184
82. ^ Kanigel 1991, p. 196
83. ^ Kanigel 1991, p. 202
84. ^ Hardy, G. H. (1940). Ramanujan. Cambridge: Cambridge University Press. p. 10.
85. ^ Letter, Littlewood to Hardy, early March 1913.
86. ^ Hardy, G. H. (1979). Collected Papers of G. H. Hardy. Oxford, England: Clarendon Press. Vol. 7, p720.
87. ^ Kanigel 1991, pp. 299–300
88. ^ Young, D. A. B. (1994). Ramanujan's illness. Notes and records of the Royal Society of London, 48(1), 107-119.
89. ^ a b c Peterson, Doug. "Raiders of the Lost Notebook". UIUC College of Liberal Arts and Sciences. Retrieved 11 January 2014.
90. ^ Gunn, J. W. C., & SAVAGE, B. (1919). Report on the Treatment of Entamoeba Histolytica Infections. Journal of the Royal Army Medical Corps, 33(5), 418-426.
91. ^
92. ^ Kanigel 1991, p. 36
93. ^ Kanigel 1991, p. 281
94. ^ Chaitin, Gregory (28 July 2007). "Less Proof, More Truth". NewScientist (2614): 49.
95. ^ Kanigel 1991, p. 283
96. ^ Berndt, Bruce C.; Rankin, Robert Alexander (2001). Ramanujan: Essays and Surveys. American Mathematical Society. p. 47. ISBN 0821826247. Retrieved 8 June 2015.
97. ^ Ranganathan 1967, p. 82
98. ^ Calyampudi Radhakrishna Rao (1997). Statistics and truth: putting chance to work. World Scientific. p. 185. ISBN 978-981-02-3111-8. Retrieved 7 June 2010.
99. ^
100. ^ a b "100-Year-Old Deathbed Dreams of Mathematician Proved True". Fox News. 28 December 2012.
101. ^ Ono (June–July 2006), p 649.
102. ^ Ramanujan, Srinivasa (1916). "On certain arithmetical functions" (PDF). Transactions of the Cambridge Philosophical Society. Cambridge Philosophical Society. XXII (No. 9). Retrieved 2016-05-15. The tau function is discussed in pages 194–197.
103. ^ Ono, Ken; Aczel, Amir D. (2016-04-13). My Search for Ramanujan: How I Learned to Count. Springer. pp. 236–237. ISBN 978-3319255668. ideas that were critical to the proof of Fermat's last theorem
104. ^ a b Ramanujans Notebooks. ASIN 0387949410.
105. ^ "Quotations by Hardy". Gap.dcs.st-and.ac.uk. Retrieved 20 November 2012.
106. ^ "Obituary Notices: Srinivasa Ramanujan". Hardy, G.H., Proceedings of the London Mathematical Society 19, p. lvii.
107. ^
108. ^ a b Srinivasa Ramanujan. Retrieved 2 December 2010.
109. ^ K Srinivasa Rao. "Srinivasa Ramanujan (22 December 1887 – 26 April 1920)".
110. ^
111. ^ "Stamps released in 1962". Indian Postage Stamps. Retrieved 22 May 2012.
112. ^ "Stamps 2011". India Post. Retrieved 22 May 2012.
113. ^ "India Post Issued a Commemorative Stamp on S Ramanujan". Phila Mirror. 26 December 2011. Retrieved 22 May 2012.
114. ^ "Singh's first visit to the state". CNN IBN. India. 26 December 2011. Retrieved 12 April 2016.
115. ^ "Racy read". 7 December 2014. Retrieved 30 April 2016.
116. ^ "Crime in a World of High Science". 16 September 2014. Retrieved 30 April 2016.
117. ^ http://www.ams.org/notices/200311/rev-ribet.pdf
118. ^ http://variety.com/2003/legit/reviews/partition-1200541626/
119. ^ http://www.thehindu.com/2003/05/26/stories/2003052604331200.htm
120. ^ http://www.thehindu.com/features/metroplus/an-ode-to-a-genius/article6708231.ece
121. ^ "'Ramanujan' Makers Shoot in His House". Indiatimes. Times Internet Limited. Retrieved 12 July 2013.
122. ^ "Camphor Cinema Presents Their First Film Ramanujan". Box Office India. Select Publishing Company. Retrieved 12 July 2013.
123. ^ "Makers of 'Ramanujan' shoot in genius' house". Z News. Zee Media Corporation Ltd. Retrieved 12 July 2013.
124. ^ Krishnamachari, Suganthy (27 June 2013). "Travails of a genius". The Hindu. Chennai, India. Retrieved 12 July 2013.
125. ^ "Cannes: Dev Patel to Star as Famed Indian Mathematician". hollywoodreporter.com.
126. ^ "Jeremy Irons to Co-star in 'The Man Who Knew Infinity'". variety.com.
127. ^
128. ^ "First Class Man". Alteregoproductions.org. Retrieved 20 November 2012.
129. ^ "News / National : James Bond director to make film on Ramanujan". The Hindu. India. 16 October 2011. Retrieved 18 October 2011.
130. ^ Nell Freudenberger (16 September 2007). "Lust for Numbers". The New York Times. Retrieved 4 September 2011.
131. ^ DJ Taylor (26 January 2008). "Adding up to a life". The Guardian. UK. Retrieved 4 September 2011.
132. ^ "Google doodles for Ramanujan's 125th birthday". Times of India. 22 December 2012. Archived from the original on 22 December 2012. Retrieved 22 December 2012.
133. ^ "Srinivasa Ramanujan's 125th Birthday". www.google.com. Retrieved 2016-04-30.
134. ^ "The Man Who Loved Numbers". Pbs.org. Retrieved 18 October 2011.