# Srinivasa Ramanujan

Indian mathematician (1887–1920)
“Ramanujan” redirects here For other uses, see Ramanujan (disambiguation)

In this Indian name, the name Srinivasa is a patronymic, and the person should be referred to by the given name, Ramanujan

Srinivasa Ramanujan

FRS
Born (1887-12-22)22 December 1887

Erode, Mysore State, British Indian Empire
Died 26 April 1920(1920-04-26) (aged 32)

Other names Srinivasa Ramanujan Aiyangar
Citizenship British India
Education
• Government Arts College (no degree)
• Pachaiyappa’s College (no degree)
• Trinity College, Cambridge (BA)
Known for
• Ramanujan’s sum
• Landau–Ramanujan constant
• Mock theta functions
• Ramanujan conjecture
• Ramanujan prime
• Ramanujan–Soldner constant
• Ramanujan theta function
• Rogers–Ramanujan identities
• Ramanujan’s master theorem
• Hardy–Ramanujan asymptotic formula
• Ramanujan–Sato series
Awards Fellow of the Royal Society (1918)
Scientific career
Fields Mathematics
Institutions Trinity College, Cambridge
Thesis Highly Composite Numbers (1916)
• G H Hardy
• J E Littlewood
Influences G S Carr
Influenced G H Hardy
Signature

Srinivasa Ramanujan FRS (/ˈsrnɪvɑːsə rɑːˈmɑːnʊən/; born Srinivasa Ramanujan Aiyangar, IPA: ; 22 December 1887 – 26 April 1920) was an Indian mathematician who lived during British Rule in India Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable Ramanujan initially developed his own mathematical research in isolation: according to Hans Eysenck: “He tried to interest the leading professional mathematicians in his work, but failed for the most part What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered” Seeking mathematicians who could better understand his work, in 1913 he began a postal correspondence with the English mathematician G H Hardy at the University of Cambridge, England Recognising Ramanujan’s work as extraordinary, Hardy arranged for him to travel to Cambridge In his notes, Hardy commented that Ramanujan had produced groundbreaking new theorems, including some that “defeated me completely; I had never seen anything in the least like them before”, and some recently proven but highly advanced results

During his short life, Ramanujan independently compiled nearly 3,900 results (mostly identities and equations) Many were completely novel; his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas of work and inspired a vast amount of further research Of his thousands of results, all but a dozen or two have now been proven correct The Ramanujan Journal, a scientific journal, was established to publish work in all areas of mathematics influenced by Ramanujan, and his notebooks—containing summaries of his published and unpublished results—have been analysed and studied for decades since his death as a source of new mathematical ideas As late as 2012, researchers continued to discover that mere comments in his writings about “simple properties” and “similar outputs” for certain findings were themselves profound and subtle number theory results that remained unsuspected until nearly a century after his death He became one of the youngest Fellows of the Royal Society and only the second Indian member, and the first Indian to be elected a Fellow of Trinity College, Cambridge Of his original letters, Hardy stated that a single look was enough to show they could have been written only by a mathematician of the highest calibre, comparing Ramanujan to mathematical geniuses such as Euler and Jacobi

In 1919, ill health—now believed to have been hepatic amoebiasis (a complication from episodes of dysentery many years previously)—compelled Ramanujan’s return to India, where he died in 1920 at the age of 32 His last letters to Hardy, written in January 1920, show that he was still continuing to produce new mathematical ideas and theorems His “lost notebook”, containing discoveries from the last year of his life, caused great excitement among mathematicians when it was rediscovered in 1976

A deeply religious Hindu, Ramanujan credited his substantial mathematical capacities to divinity, and said his family goddess, Namagiri Thayar, revealed his mathematical knowledge to him He once said, “An equation for me has no meaning unless it expresses a thought of God”

## Early life

Ramanujan’s birthplace on 18 Alahiri Street, Erode, now in Tamil Nadu

Ramanujan’s home on Sarangapani Sannidhi Street, Kumbakonam

Ramanujan (literally, “younger brother of Rama”, a Hindu deity was born on 22 December 1887 into a Tamil Brahmin Iyengar family in Erode during the British Colonial Rule (present-day Tamil Nadu, India), at the residence of his maternal grandparents His father, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, worked as a clerk in a sari shop His mother, Komalatammal, was a housewife and sang at a local temple They lived in a small traditional home on Sarangapani Sannidhi Street 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, Sadagopan, who died less than three months later In December 1889, Ramanujan contracted smallpox, but recovered, unlike the 4,000 others who died in a bad year in the Thanjavur district around this time 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, both of whom died before their first birthdays

On 1 October 1892, Ramanujan was enrolled at the local school After his maternal grandfather lost his job as a court official in Kanchipuram, Ramanujan and his mother moved back to Kumbakonam, and he was enrolled in Kangayan Primary School 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 he attended school Within six months, Ramanujan was back in Kumbakonam

Since Ramanujan’s father was at work most of the day, his mother took care of the boy, and they had a close relationship From her, he learned about tradition and puranas, to sing religious songs, to attend pujas at the temple, and to maintain particular eating habits—all part of Brahmin culture At 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 That year, Ramanujan entered Town Higher Secondary School, where he encountered formal mathematics for the first time

A child prodigy 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 written by S L Loney on advanced trigonometry He mastered this by the age of 13 while discovering sophisticated theorems on his own By 14, he received merit certificates and academic awards that continued throughout his school career, and he assisted the school in the logistics of assigning its 1,200 students (each with differing needs) to its approximately 35 teachers 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 would later develop his own method to solve the quartic In 1903, he tried to solve the quintic, not knowing that it was impossible to solve with radicals

In 1903, when he was 16, Ramanujan obtained from a friend a library copy of A Synopsis of Elementary Results in Pure and Applied Mathematics, G S Carr’s collection of 5,000 theorems Ramanujan reportedly studied the contents of the book in detail The next year, Ramanujan independently developed and investigated the Bernoulli numbers and calculated the Euler–Mascheroni constant up to 15 decimal places His peers at the time said they “rarely understood him” and “stood in respectful awe” of him

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 He received a scholarship to study at Government Arts College, Kumbakonam, 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 In August 1905, Ramanujan ran away from home, heading towards Visakhapatnam, and stayed in Rajahmundry for about a month 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 Ramanujan failed his Fellow of Arts exam in December 1906 and again a year later Without an FA degree, he left college and continued to pursue independent research in mathematics, living in extreme poverty and often on the brink of starvation

In 1910, after a meeting between the 23-year-old Ramanujan and the founder of the Indian Mathematical Society, V Ramaswamy Aiyer, Ramanujan began to get recognition in Madras’s mathematical circles, leading to his inclusion as a researcher at the University of Madras

On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 March 1899 – 13 April 1994), a girl his mother had selected for him a year earlier and who was ten years old when they married It was not unusual then for marriages to be arranged with girls at a young age Janaki was from Rajendram, a village close to Marudur (Karur district) Railway Station Ramanujan’s father did not participate in the marriage ceremony As was common at that time, Janaki continued to stay at her maternal home for three years after marriage, until she reached puberty In 1912, she and Ramanujan’s mother joined Ramanujan in Madras

After the marriage, Ramanujan developed a hydrocele testis The condition could be treated with a routine surgical operation that would release the blocked fluid in the scrotal sac, but his family could not afford the operation In January 1910, a doctor volunteered to do the surgery at no cost

After his successful surgery, Ramanujan searched for a job He stayed at a friend’s 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 Fellow of Arts exam

In late 1910, Ramanujan was sick again He feared for his health, and told his friend R Radakrishna Iyer to “hand over to Professor Singaravelu Mudaliar or to the British professor Edward B Ross, of the Madras Christian College” After Ramanujan recovered and retrieved his notebooks from Iyer, he took a train from Kumbakonam to Villupuram, a city under French control In 1912, Ramanujan moved with his wife and mother to a house in Saiva Muthaiah Mudali street, George Town, Madras, where they lived for a few months In May 1913, upon securing a research position at Madras University, Ramanujan moved with his family to Triplicane

### Pursuit of career in mathematics

In 1910, Ramanujan met deputy collector V Ramaswamy Aiyer, who founded the Indian Mathematical Society Wishing for a job at the revenue department where Aiyer worked, Ramanujan showed him his mathematics notebooks As Aiyer later recalled:

Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras 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 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 fraud 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 convinced him of Ramanujan’s brilliance 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 With Aiyer’s help, Ramanujan had his work published in the Journal of the Indian Mathematical Society

One of the first problems he posed in the journal was to find the value of:

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

He waited for a solution to be offered in three issues, over six months, but failed to receive any At the end, Ramanujan supplied an incomplete 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

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

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 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 are 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. 2(2 − 1)Bn/n is an integer and 2(2 − 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 His writing initially had many flaws As Journal editor M T Narayana Iyengar noted:

Ramanujan later wrote another paper and also continued to provide problems in the Journal In early 1912, he got a temporary job in the Madras Accountant General’s office, with a monthly salary of 20 rupees He lasted only a few weeks 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:

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” Three weeks after he applied, on 1 March, Ramanujan learned that he had been accepted as a Class III, Grade IV accounting clerk, making 30 rupees per month At his office, Ramanujan easily and quickly completed the work he was given and 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 He said that although Ramanujan had “a taste for mathematics, and some ability”, he lacked the necessary educational background and foundation to be accepted by mathematicians Although Hill did not offer to take Ramanujan on as a student, he gave thorough and serious professional advice on his work With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University

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

$\underset{0}{\overset{\infty }{\int }}\frac{1+\frac{{x}^{2}}{\left(b+1{\right)}^{2}}}{1+\frac{{x}^{2}}{{a}^{2}}}×\frac{1+\frac{{x}^{2}}{\left(b+2{\right)}^{2}}}{1+\frac{{x}^{2}}{\left(a+1{\right)}^{2}}}×\cdots \phantom{\rule{0ex}{0ex}}dx=\frac{\sqrt{\pi }}{2}×\frac{\Gamma \left(a+\frac{1}{2}\right)\Gamma \left(b+1\right)\Gamma \left(b-a+1\right)}{\Gamma \left(a\right)\Gamma \left(b+\frac{1}{2}\right)\Gamma \left(b-a+\frac{1}{2}\right)}$

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

$1-5{\left(\frac{1}{2}\right)}^{3}+9{\left(\frac{1×3}{2×4}\right)}^{3}-13{\left(\frac{1×3×5}{2×4×6}\right)}^{3}+\cdots =\frac{2}{\pi }$

$1+9{\left(\frac{1}{4}\right)}^{4}+17{\left(\frac{1×5}{4×8}\right)}^{4}+25{\left(\frac{1×5×9}{4×8×12}\right)}^{4}+\cdots =\frac{2\sqrt{2}}{\sqrt{\pi }\phantom{\rule{0ex}{0ex}}{\Gamma }^{2}\left(\frac{3}{4}\right)}$

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 Euler and Gauss Hardy found these results “much more intriguing” than Gauss’s work on integrals After seeing Ramanujan’s theorems on continued fractions on the last page of the manuscripts, Hardy said the theorems “defeated me completely; I had never seen anything in the least like them before”, and that they “must be true, because, if they were not true, no one would have the imagination to invent them” 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 that Ramanujan was “a mathematician of the highest quality, a man of altogether exceptional originality and power” One colleague, E H Neville, later remarked that “not one could have been set in the most advanced mathematical examination in the world”

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

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 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” The board agreed to grant Ramanujan a monthly research scholarship of 75 rupees for the next two years at the University of Madras

While he was engaged as a research student, Ramanujan continued to submit papers to the Journal of the Indian Mathematical Society In one instance, 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 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

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 Neville asked Ramanujan why he would not go to Cambridge Ramanujan apparently had now accepted the proposal; Neville said, “Ramanujan needed no converting” and “his parents’ opposition had been withdrawn” 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” On 17 March 1914, Ramanujan traveled to England by ship, leaving his wife to stay with his parents in India

## Life in England

Ramanujan (centre) and his colleague G H Hardy (rightmost), with other scientists, outside the Senate House, Cambridge, c1914–19

Whewell’s Court, Trinity College, Cambridge

Ramanujan departed from Madras aboard the SS Nevasa on 17 March 1914 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

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 Ramanujan left a deep impression on Hardy and Littlewood Littlewood commented, “I can believe that he’s at least a Jacobi”, while Hardy said he “can compare him only with Euler or Jacobi”

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 In the previous few decades, the foundations of mathematics had come into question and the need for mathematically rigorous proofs recognised 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 and insights Hardy tried his best to fill the gaps in Ramanujan’s education and to mentor him in the need for formal proofs to support his results, without hindering his inspiration—a conflict that neither found easy

Ramanujan was awarded a Bachelor of Arts by Research degree (the predecessor of the PhD degree) in March 1916 for his work on highly composite numbers, sections of the first part of which had been published the preceding year in the Proceedings of the London Mathematical Society The paper was more than 50 pages long and proved various properties of such numbers Hardy disliked this topic area but remarked that though it engaged with what he called the ‘backwater of mathematics’, in it Ramanujan displayed ‘extraordinary mastery over the algebra of inequalities’

On 6 December 1917, Ramanujan was elected to the London Mathematical Society On 2 May 1918, he was elected a Fellow of the Royal Society, the second Indian admitted, after Ardaseer Cursetjee in 1841 At age 31, Ramanujan was one of the youngest Fellows in the Royal Society’s history 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

## Illness and death

Ramanujan had numerous health problems throughout his life His health worsened in England; possibly he was also less resilient due to the difficulty of keeping to the strict dietary requirements of his religion there and because of wartime rationing in 1914–18 He was diagnosed with tuberculosis and a severe vitamin deficiency, and confined to a sanatorium In 1919, he returned to Kumbakonam, Madras Presidency, and in 1920 he died at the age of 32 After his death, his brother Tirunarayanan compiled Ramanujan’s remaining handwritten notes, consisting of formulae on singular moduli, hypergeometric series and continued fractions

Ramanujan’s widow, Smt Janaki Ammal, moved to Bombay In 1931, she returned to Madras and settled in Triplicane, where she supported herself on a pension from Madras University and income from tailoring In 1950, she adopted a son, W Narayanan, who eventually became an officer of the State Bank of India and raised a family In her later years, she was granted a lifetime pension from Ramanujan’s former employer, the Madras Port Trust, and pensions from, among others, the Indian National Science Academy and the state governments of Tamil Nadu, Andhra Pradesh and West Bengal She continued to cherish Ramanujan’s memory, and was active in efforts to increase his public recognition; prominent mathematicians, including George Andrews, Bruce C Berndt and Béla Bollobás made it a point to visit her while in India She died at her Triplicane residence in 1994

A 1994 analysis of Ramanujan’s medical records and symptoms by Dr D A B Young concluded that his medical symptoms—including his past relapses, fevers, and hepatic conditions—were much closer to those resulting from hepatic amoebiasis, an illness then widespread in Madras, than tuberculosis He had two episodes of dysentery before he left India When not properly treated, amoebic dysentery can lie dormant for years and lead to hepatic amoebiasis, whose diagnosis was not then well established At the time, if properly diagnosed, amoebiasis was a treatable and often curable disease; British soldiers who contracted it during the First World War were being successfully cured of amoebiasis around the time Ramanujan left England

## Personality and spiritual life

Ramanujan has been described as a person of a somewhat shy and quiet disposition, a dignified man with pleasant manners He lived a simple life at Cambridge Ramanujan’s first Indian biographers describe him as a rigorously orthodox Hindu He credited his acumen to his family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal He looked to her for inspiration in his work and said he dreamed of blood drops that symbolised her consort, Narasimha Later he had visions of scrolls of complex mathematical content unfolding before his eyes He often said, “An equation for me has no meaning unless it expresses a thought of God”

Hardy cites Ramanujan as remarking that all religions seemed equally true to him 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

Similarly, in an interview with Frontline, Berndt said, “Many people falsely promulgate mystical powers to Ramanujan’s mathematical thinking It is not true He has meticulously recorded every result in his three notebooks,” further speculating that Ramanujan worked out intermediate results on slate that he could not afford the paper to record more permanently

## Mathematical achievements

In mathematics, there is a distinction between insight and formulating or working through 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 intriguing of these formulae include infinite series for π, one of which is given below:

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

This result is based on the negative fundamental discriminant d = −4 × 58 = −232 with class number h(d) = 2 Further, 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 396, which is related to the fact that

${e}^{\pi \sqrt{58}}={396}^{4}-104000000177\dots$

This might be compared to Heegner numbers, which have class number 1 and yield similar formulae

Ramanujan’s series for π converges extraordinarily rapidly 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; truncating it to the first two terms gives a value correct to 14 decimal places See also the more general Ramanujan–Sato series

One of Ramanujan’s remarkable capabilities was the rapid solution of problems, illustrated by the following anecdote about an incident in which P C Mahalanobis posed a problem:

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

$\begin{array}{cc}& {\left(1+2\sum _{n=1}^{\infty }\frac{\mathrm{cos}\left(n\theta \right)}{\mathrm{cosh}\left(n\pi \right)}\right)}^{-2}+{\left(1+2\sum _{n=1}^{\infty }\frac{\mathrm{cosh}\left(n\theta \right)}{\mathrm{cosh}\left(n\pi \right)}\right)}^{-2}\\ =& \frac{2{\Gamma }^{4}\left(\frac{3}{4}\right)}{\pi }=\frac{8{\pi }^{3}}{{\Gamma }^{4}\left(\frac{1}{4}\right)}\end{array}$

for all θ such that

$|\Re \left(\theta \right)|<\pi$

and

$|\Im \left(\theta \right)|<\pi$

, 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 θ, θ, and θ 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 In 1937, Hans Rademacher refined 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

In the last year of his life, Ramanujan discovered mock theta functions 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, one 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

In his paper “On certain arithmetical functions”, Ramanujan defined the so-called delta-function, whose coefficients are called τ(n) (the Ramanujan tau function) He proved many congruences for these numbers, such as τ(p) ≡ 1 + p 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 that “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 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

### Ramanujan’s notebooks

Further information: Ramanujan’s lost notebook

While still in Madras, Ramanujan recorded the bulk of his results in four notebooks of looseleaf paper They were mostly written up without any derivations This is probably the origin of the misapprehension 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 record the proofs in his notes

This may have been for any number of reasons Since paper was very expensive, Ramanujan did most of his work and perhaps his proofs on slate, after which he transferred the final results to paper At the time, slates were commonly used by mathematics students in the Madras Presidency He was also quite likely to have been influenced by the style of G S Carr’s book, which stated results without proofs It is also possible that Ramanujan considered his work to be for his personal interest alone and therefore recorded only the results

The first notebook has 351 pages with 16 somewhat organised chapters and some unorganised material The second has 256 pages in 21 chapters and 100 unorganised pages, and the third 33 unorganised pages The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found Hardy himself wrote papers exploring material from Ramanujan’s work, as did G N Watson, B M Wilson, and Bruce Berndt

In 1976, George Andrews rediscovered a fourth notebook with 87 unorganised pages, the so-called “lost notebook”

## 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:

Immediately before this anecdote, Hardy quoted Littlewood as saying, “Every positive integer was one of personal friends”

The two different ways are:

$1729={1}^{3}+{12}^{3}={9}^{3}+{10}^{3}$

Generalisations of this idea have created the notion of “taxicab numbers”

## Mathematicians’ views of Ramanujan

In his obituary of Ramanujan, written for Nature in 1920, Hardy observed that Ramanujan’s work primarily involved fields less known even among other pure mathematicians, concluding:

Hardy further said:

When asked about the methods Ramanujan employed to arrive at his solutions, Hardy said they were “arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account” He also said that he had “never met his equal, and can compare him only with Euler or Jacobi”

K Srinivasa Rao has said, “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 May 2011 lecture at IIT Madras, Berndt said that over the last 40 years, as nearly all of Ramanujan’s conjectures had been proven, 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

## Posthumous recognition

Further information: List of things named after Srinivasa Ramanujan

Bust of Ramanujan in the garden of Birla Industrial & Technological Museum in Kolkata, India

The year after his death, Nature listed Ramanujan among other distinguished scientists and mathematicians on a “Calendar of Scientific Pioneers” who had achieved eminence Ramanujan’s home state of Tamil Nadu celebrates 22 December (Ramanujan’s birthday) as ‘State IT Day’ Stamps picturing Ramanujan were issued by the government of India in 1962, 2011, 2012 and 2016

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 The International Centre for Theoretical Physics (ICTP) has created a prize in Ramanujan’s name for young mathematicians from developing countries in cooperation with the International Mathematical Union, which nominates members of the prize committee SASTRA University, a private university based in Tamil Nadu, has instituted the SASTRA Ramanujan Prize of US$10,000 to be given annually to a mathematician not exceeding age 32 for outstanding contributions in an area of mathematics influenced by Ramanujan Based on the recommendations of a committee appointed by the University Grants Commission (UGC), Government of India, the Srinivasa Ramanujan Centre, established by SASTRA, has been declared an off-campus centre under the ambit of SASTRA University House of Ramanujan Mathematics, a museum of Ramanujan’s life and work, is also on this campus SASTRA purchased and renovated the house where Ramanujan lived at Kumabakonam 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 Then Indian Prime Minister Manmohan Singh also declared that 2012 would be celebrated as National Mathematics Year Ramanujan IT City is an information technology (IT) special economic zone (SEZ) in Chennai that was built in 2011 Situated next to the Tidel Park, it includes 25 acres (10 ha) with two zones, with a total area of 57 million square feet (530,000 m), including 45 million square feet (420,000 m) of office space ## Commemorative postal stamps Commemorative stamps released by India Post (by year): 1962 2011 2012 2016 ## In popular culture • The Man Who Loved Numbers is a 1988 PBS NOVA documentary about Ramanujan (S15, E9) • The Man Who Knew Infinity is a 2015 film based on Kanigel’s book British actor Dev Patel portrays Ramanujan • Ramanujan, an Indo-British collaboration film chronicling Ramanujan’s life, was released in 2014 by the independent film company Camphor Cinema The cast and crew include director Gnana Rajasekaran, cinematographer Sunny Joseph and editor B Lenin Indian and English stars Abhinay Vaddi, Suhasini Maniratnam, Bhama, Kevin McGowan and Michael Lieber star in pivotal roles • Nandan Kudhyadi directed the Indian documentary films The Genius of Srinivasa Ramanujan (2013) and Srinivasa Ramanujan: The Mathematician and His Legacy (2016) about the mathematician • Ramanujan (The Man Who Reshaped 20th Century Mathematics), an Indian docudrama film directed by Akashdeep released in 2018 • M N Krish’s thriller novel The Steradian Trail weaves Ramanujan and his accidental discovery into its plot connecting religion, mathematics, finance and economics • Partition, a play by Ira Hauptman about Hardy and Ramanujan, was first performed in 2013 • The play First Class Man by Alter Ego Productions was based on David Freeman’s First Class Man The play centres around Ramanujan and his complex and dysfunctional relationship with Hardy On 16 October 2011 it was announced that Roger Spottiswoode, best known for his James Bond film Tomorrow Never Dies, is working on the film version, starring Siddharth • A Disappearing Number is a British stage production by the company Complicite that explores the relationship between Hardy and Ramanujan • David Leavitt’s novel The Indian Clerk explores the events following Ramanujan’s letter to Hardy • Google honoured Ramanujan on his 125th birth anniversary by replacing its logo with a doodle on its home page • Ramanujan was mentioned in the 1997 film Good Will Hunting, in a scene where professor Gerald Lambeau (Stellan Skarsgård) explains to Sean Maguire (Robin Williams) the genius of Will Hunting (Matt Damon) by comparing him to Ramanujan ## Selected papers • Ramanujan, S (1914) “Some definite integrals” Messenger Math 44: 10–18 • Ramanujan, S (1914) “Some definite integrals connected with Gauss’s sums” Messenger Math 44: 75–85 • Ramanujan, S (1915) “On certain infinite series” Messenger Math 45: 11–15 • Ramanujan, S (1915) “Highly Composite Numbers” Proceedings of the London Mathematical Society 14 (1): 347–409 doi:101112/plms/s2_141347 • Ramanujan, S (1915) “On the number of divisors of a number” The Journal of the Indian Mathematical Society 7 (4): 131–133 • Ramanujan, S (1915) “Short Note: On the sum of the square roots of the first n natural numbers” The Journal of the Indian Mathematical Society 7 (5): 173–175 • Ramanujan, S (1916) “Some formulae in the analytical theory of numbers” Messenger Math 45: 81–84 • Ramanujan, S (1916) “A Series for Euler’s Constant γ” Messenger Math 46: 73–80 • Ramanujan, S (1917) “On the expression of numbers in the form ax + by + cz + duMathematical Proceedings of the Cambridge Philosophical Society 19: 11–21 • Hardy, G H; Ramanujan, S (1917) “Asymptotic Formulae for the Distribution of Integers of Various Types” Proceedings of the London Mathematical Society 16 (1): 112–132 doi:101112/plms/s2-161112 • Hardy, G H; Ramanujan, Srinivasa (1918) “Asymptotic Formulae in Combinatory Analysis” Proceedings of the London Mathematical Society 17 (1): 75–115 doi:101112/plms/s2-17175 • Hardy, G H; Ramanujan, Srinivasa (1918) “On the coefficients in the expansions of certain modular functions” Proc R Soc A 95 (667): 144–155 Bibcode:1918RSPSA95144H doi:101098/rspa19180056 • Ramanujan, Srinivasa (1919) “Some definite integrals” The Journal of the Indian Mathematical Society 11 (2): 81–88 • Ramanujan, S (1919) “A proof of Bertrand’s postulate” The Journal of the Indian Mathematical Society 11 (5): 181–183 • Ramanujan, S (1920) “A class of definite integrals” Quart J Pure Appl Math 48: 294–309 hdl:2027/uc1$b417568
• Ramanujan, S (1921) “Congruence properties of partitions” Math Z 9 (1–2): 147–153 doi:101007/BF01378341 S2CID 121753215 Posthumously published extract of a longer, unpublished manuscript

## Further works of Ramanujan’s mathematics

• George E Andrews and Bruce C Berndt, Ramanujan’s Lost Notebook: Part I (Springer, 2005, ISBN 0-387-25529-X)
• George E Andrews and Bruce C Berndt, Ramanujan’s Lost Notebook: Part II, (Springer, 2008, ISBN 978-0-387-77765-8)
• George E Andrews and Bruce C Berndt, Ramanujan’s Lost Notebook: Part III, (Springer, 2012, ISBN 978-1-4614-3809-0)
• George E Andrews and Bruce C Berndt, Ramanujan’s Lost Notebook: Part IV, (Springer, 2013, ISBN 978-1-4614-4080-2)
• George E Andrews and Bruce C Berndt, Ramanujan’s Lost Notebook: Part V, (Springer, 2018, ISBN 978-3-319-77832-7)
• M P Chaudhary, A simple solution of some integrals given by Srinivasa Ramanujan, (Resonance: J Sci Education – publication of Indian Academy of Science, 2008)
• MP Chaudhary, Mock theta functions to mock theta conjectures, SCIENTIA, Series A : Math Sci, (22)(2012) 33–46
• MP Chaudhary, On modular relations for the Roger-Ramanujan type identities, Pacific J Appl Math, 7(3)(2016) 177–184

## Selected publications on Ramanujan and his work

• Berndt, Bruce C (1998) Butzer, P L; Oberschelp, W; Jongen, H Th (eds) Charlemagne and His Heritage: 1200 Years of Civilization and Science in Europe (PDF) Turnhout, Belgium: Brepols Verlag pp 119–146 ISBN 978-2-503-50673-9
• Berndt, Bruce C; Rankin, Robert A (1995) Ramanujan: Letters and Commentary Vol 9 Providence, Rhode Island: American Mathematical Society ISBN 978-0-8218-0287-8
• Berndt, Bruce C; Rankin, Robert A (2001) Ramanujan: Essays and Surveys Vol 22 Providence, Rhode Island: American Mathematical Society ISBN 978-0-8218-2624-9
• Berndt, Bruce C (2006) Number Theory in the Spirit of Ramanujan Vol 9 Providence, Rhode Island: American Mathematical Society ISBN 978-0-8218-4178-5
• Berndt, Bruce C (1985) Ramanujan’s Notebooks: Part I New York: Springer ISBN 978-0-387-96110-1
• Berndt, Bruce C (1999) Ramanujan’s Notebooks: Part II New York: Springer ISBN 978-0-387-96794-3
• Berndt, Bruce C (2004) Ramanujan’s Notebooks: Part III New York: Springer ISBN 978-0-387-97503-0
• Berndt, Bruce C (1993) Ramanujan’s Notebooks: Part IV New York: Springer ISBN 978-0-387-94109-7
• Berndt, Bruce C (2005) Ramanujan’s Notebooks: Part V New York: Springer ISBN 978-0-387-94941-3
• Hardy, G H (March 1937) “The Indian Mathematician Ramanujan” The American Mathematical Monthly 44 (3): 137–155 doi:102307/2301659 JSTOR 2301659
• Hardy, G H (1978) Ramanujan New York: Chelsea Pub Co ISBN 978-0-8284-0136-4
• Hardy, G H (1999) Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work Providence, Rhode Island: American Mathematical Society ISBN 978-0-8218-2023-0
• Henderson, Harry (1995) Modern Mathematicians New York: Facts on File Inc ISBN 978-0-8160-3235-8
• Kanigel, Robert (1991) The Man Who Knew Infinity: a Life of the Genius Ramanujan New York: Charles Scribner’s Sons ISBN 978-0-684-19259-8
• Leavitt, David (2007) The Indian Clerk (paperback ed) London: Bloomsbury ISBN 978-0-7475-9370-6
• Narlikar, Jayant V (2003) Scientific Edge: the Indian Scientist From Vedic to Modern Times New Delhi, India: Penguin Books ISBN 978-0-14-303028-7
• Ono, Ken; Aczel, Amir D (13 April 2016) My Search for Ramanujan: How I Learned to Count Springer ISBN 978-3319255668
• Sankaran, T M (2005) “Srinivasa Ramanujan- Ganitha lokathile Mahaprathibha” (in Malayalam) Kochi, India: Kerala Sastra Sahithya Parishath {{cite journal}}: Cite journal requires |journal= (help)

## 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 978-0-8218-2076-6
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 978-3-540-18726-4
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