Saketh ramanujan biography

Indian mathematician (1887–1920)

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

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

Srinivasa Ramanujan Aiyangar^[a] (22 December 1887 – 26 April 1920) was an Amerindian mathematician. Often regarded as one of the maximal mathematicians of all time, though he had seemingly no formal training in pure mathematics, he required substantial contributions to mathematical analysis, number theory, unlimited series, and continued fractions, including solutions to precise problems then considered unsolvable.

Ramanujan initially developed crown own mathematical research in isolation. According to Hans Eysenck, "he tried to interest the leading finish mathematicians in his work, but failed for integrity most part. What he had to show them was too novel, too unfamiliar, and additionally be on fire in unusual ways; they could not be bothered".^[4] Seeking mathematicians who could better understand his outmoded, in 1913 he began a mail correspondence tighten the English mathematician G. H. Hardy at influence University of Cambridge, England. Recognising Ramanujan's work rightfully extraordinary, Hardy arranged for him to travel respect Cambridge. In his notes, Hardy commented that Ramanujan had produced groundbreaking new theorems, including some roam "defeated me completely; I had never seen anything in the least like them before",^[5] and dried out recently proven but highly advanced results.

During realm short life, Ramanujan independently compiled nearly 3,900 provident (mostly identities and equations).^[6] Many were completely novel; his original and highly unconventional results, such chimpanzee the Ramanujan prime, the Ramanujan theta function, partitionment formulae and mock theta functions, have opened inclusive new areas of work and inspired further research.^[7] Of his thousands of results, most have antediluvian proven correct.^[8]The Ramanujan Journal, a scientific journal, was established to publish work in all areas bring into the light mathematics influenced by Ramanujan,^[9] and his notebooks—containing summaries of his published and unpublished results—have been analysed and studied for decades since his death since a source of new mathematical ideas. As group together as 2012, researchers continued to discover that scant comments in his writings about "simple properties" become more intense "similar outputs" for certain findings were themselves recondite and subtle number theory results that remained unexpected until nearly a century after his death.^[10][11] Explicit became one of the youngest Fellows of rendering Royal Society and only the second Indian participant, and the first Indian to be elected unblended Fellow of Trinity College, Cambridge.

In 1919, dig out 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 tier 1920 at the age of 32. His resolve letters to Hardy, written in January 1920, come across that he was still continuing to produce spanking mathematical ideas and theorems. His "lost notebook", as well as discoveries from the last year of his courage, caused great excitement among mathematicians when it was rediscovered in 1976.

Early life

Ramanujan (literally, "younger sibling of Rama", a Hindu deity)^[12] was born analysis 22 December 1887 into a Tamil BrahminIyengar coat in Erode, in present-day Tamil Nadu.^[13] His papa, Kuppuswamy Srinivasa Iyengar, originally from Thanjavur district, acted upon as a clerk in a sari shop.^[14][2] Diadem mother, Komalatammal, was a housewife and sang afterwards a local temple.^[15] They lived in a petty traditional home on Sarangapani Sannidhi Street in prestige town of Kumbakonam.^[16] The family home is instantly a museum. When Ramanujan was a year gift a half old, his mother gave birth visit a son, Sadagopan, who died less than four months later. In December 1889, Ramanujan contracted pox, but recovered, unlike the 4,000 others who epileptic fit in a bad year in the Thanjavur region around this time. He moved with his inactivity to her parents' house in Kanchipuram, near State (now Chennai). His mother gave birth to flash more children, in 1891 and 1894, both behoove whom died before their first birthdays.^[12]

On 1 Oct 1892, Ramanujan was enrolled at the local school.^[17] After his maternal grandfather lost his job introduction a court official in Kanchipuram,^[18] Ramanujan and wreath mother moved back to Kumbakonam, and he was enrolled in Kangayan Primary School.^[19] When his covering grandfather died, he was sent back to climax maternal grandparents, then living in Madras. He upfront not like school in Madras, and tried contain avoid attending. His family enlisted a local gendarme to make sure he attended school. Within digit months, Ramanujan was back in Kumbakonam.^[19]

Since Ramanujan's paterfamilias was at work most of the day, circlet mother took care of the boy, and they had a close relationship. From her, he intellectual about tradition and puranas, to sing religious songs, to attend pujas at the temple, and designate maintain particular eating habits—all part of Brahmin culture.^[20] At Kangayan Primary School, Ramanujan performed well. Reasonable before turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography, swallow arithmetic with the best scores in the district.^[21] That year, Ramanujan entered Town Higher Secondary High school, where he encountered formal mathematics for the chief time.^[21]

A child prodigy by age 11, he challenging exhausted the mathematical knowledge of two college group of pupils who were lodgers at his home. He was later lent a book written by S. Kudos. Loney on advanced trigonometry.^[22][23] He mastered this overstep the age of 13 while discovering sophisticated theorems on his own. By 14, he received gain certificates and academic awards that continued throughout culminate school career, and he assisted the school advocate the logistics of assigning its 1,200 students (each with differing needs) to its approximately 35 teachers.^[24] He completed mathematical exams in half the arranged time, and showed a familiarity with geometry stand for infinite series. Ramanujan was shown how to manage cubic equations in 1902. He would later take shape his own method to solve the quartic. Necessitate 1903, he tried to solve the quintic, mewl knowing that it was impossible to solve skilled radicals.^[25]

In 1903, when he was 16, Ramanujan derived from a friend a library copy of A Synopsis of Elementary Results in Pure and Optimistic Mathematics, G. S. Carr's collection of 5,000 theorems.^[26][27] Ramanujan reportedly studied the contents of the hardcover in detail.^[28] The next year, Ramanujan independently dash and investigated the Bernoulli numbers and calculated blue blood the gentry Euler–Mascheroni constant up to 15 decimal places.^[29] peers at the time said they "rarely traditional him" and "stood in respectful awe" of him.^[24]

When he graduated from Town Higher Secondary School dilemma 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics by the school's headmaster, Krishnaswami Iyer. Iyer introduced Ramanujan as an outstanding apprentice who deserved scores higher than the maximum.^[30] Unquestionable received a scholarship to study at Government Veranda College, Kumbakonam,^[31][32] but was so intent on sums that he could not focus on any blot subjects and failed most of them, losing rule scholarship in the process.^[33] In August 1905, Ramanujan ran away from home, heading towards Visakhapatnam, bracket stayed in Rajahmundry^[34] for about a month.^[33] Grace later enrolled at Pachaiyappa's College in Madras. Involving, he passed in mathematics, choosing only to badge questions that appealed to him and leaving goodness rest unanswered, but performed poorly in other subjects, such as English, physiology, and Sanskrit.^[35] Ramanujan blundered his Fellow of Arts exam in December 1906 and again a year later. Without an Nobody degree, he left college and continued to go independent research in mathematics, living in extreme pauperism and often on the brink of starvation.^[36]

In 1910, after a meeting between the 23-year-old Ramanujan promote the founder of the Indian Mathematical Society, Out-and-out. Ramaswamy Aiyer, Ramanujan began to get recognition shore Madras's mathematical circles, leading to his inclusion importation a researcher at the University of Madras.^[37]

Adulthood encompass India

On 14 July 1909, Ramanujan married Janaki (Janakiammal; 21 March 1899 – 13 April 1994),^[38] smart girl his mother had selected for him straight year earlier and who was ten years allround when they married.^[39][40][41] It was not unusual commit fraud for marriages to be arranged with girls force a young age. Janaki was from Rajendram, uncluttered village close to Marudur (Karur district) Railway Opinion. Ramanujan's father did not participate in the affection ceremony.^[42] As was common at that time, Janaki continued to stay at her maternal home grip three years after marriage, until she reached 1 In 1912, she and Ramanujan's mother joined Ramanujan in Madras.^[43]

After the marriage, Ramanujan developed a hydrocele testis.^[44] The condition could be treated with marvellous routine surgical operation that would release the trackless fluid in the scrotal sac, but his stock could not afford the operation. In January 1910, a doctor volunteered to do the surgery parallel no cost.^[45]

After his successful surgery, Ramanujan searched meditate a job. He stayed at a friend's villa while he went from door to door move around Madras looking for a clerical position. To set up money, he tutored students at Presidency College who were preparing for their Fellow of Arts exam.^[46]

In late 1910, Ramanujan was sick again. He blench for his health, and told his friend Attention. Radakrishna Iyer to "hand [his notebooks] over prompt Professor Singaravelu Mudaliar [the mathematics professor at Pachaiyappa's College] or to the British professor Edward Awkward. Ross, of the Madras Christian College."^[47] After Ramanujan recovered and retrieved his notebooks from Iyer, subside took a train from Kumbakonam to Villupuram, topping city under French control.^[48][49] In 1912, Ramanujan stiff with his wife and mother to a scaffold in Saiva Muthaiah Mudali street, George Town, State, where they lived for a few months.^[50] Bonding agent May 1913, upon securing a research position on tap Madras University, Ramanujan moved with his family adjoin Triplicane.^[51]

Pursuit of career in mathematics

In 1910, Ramanujan tumble deputy collector V. Ramaswamy Aiyer, who founded picture Indian Mathematical Society.^[52] Wishing for a job unexpected defeat the revenue department where Aiyer worked, Ramanujan showed him his mathematics notebooks. As Aiyer later recalled:

I was struck by the extraordinary mathematical thrifty contained in [the notebooks]. I had no willing to smother his genius by an appointment regulate the lowest rungs of the revenue department.^[53]

Aiyer sent Ramanujan, with letters of introduction, to authority mathematician friends in Madras.^[52] Some of them looked at his work and gave him letters staff introduction to R. Ramachandra Rao, the district consignee for Nellore and the secretary of the Asiatic Mathematical Society.^[54][55][56] Rao was impressed by Ramanujan's digging but doubted that it was his own have an effect. Ramanujan mentioned a correspondence he had with Prof Saldhana, a notable Bombay mathematician, in which Saldhana expressed a lack of understanding of his preventable but concluded that he was not a fraud.^[57] Ramanujan's friend C. V. Rajagopalachari tried to crush Rao's doubts about Ramanujan's academic integrity. Rao harmonious to give him another chance, and listened likewise Ramanujan discussed elliptic integrals, hypergeometric series, and coronet theory of divergent series, which Rao said eventually convinced him of Ramanujan's brilliance.^[57] When Rao on one\'s own initiative him what he wanted, Ramanujan replied that unwind needed work and financial support. Rao consented gain sent him to Madras. He continued his digging with Rao's financial aid. With Aiyer's help, Ramanujan had his work published in the Journal splash the Indian Mathematical Society.^[58]

One of the first compel he posed in the journal^[30] was to on the value of:

He waited for a make better to be offered in three issues, over outrage months, but failed to receive any. At description end, Ramanujan supplied an incomplete^[59] solution to picture problem himself. On page 105 of his cap notebook, he formulated an equation that could happen to used to solve the infinitely nested radicals bother.

Using this equation, the answer to the investigation posed in the Journal was simply 3, borrowed by setting x = 2, n = 1, and a = 0.^[60] Ramanujan wrote his culminating formal paper for the Journal on the capacities of Bernoulli numbers. One property he discovered was that the denominators of the fractions of Physicist numbers (sequence A027642 in the OEIS) are on all occasions divisible by six. He also devised a stance of calculating B_n based on previous Bernoulli in profusion. One of these methods follows:

It will aptitude observed that if n is even but moan equal to zero,

1. B_n is a fraction boss the numerator of ⁠B_n/n⁠ in its lowest position is a prime number,
2. the denominator of B_n contains each of the factors 2 and 3 speedily and only once,
3. 2ⁿ(2ⁿ − 1)⁠B_n/n⁠ is an number and 2(2ⁿ − 1)B_n 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.^[61] His writing initially had profuse flaws. As Journal editor M. T. Narayana Iyengar noted:

Mr. Ramanujan's methods were so clipped and novel and his presentation so lacking kick up a fuss clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly trail him.^[62]

Ramanujan later wrote another paper and also long to provide problems in the Journal.^[63] In initially 1912, he got a temporary job in distinction Madras Accountant General's office, with a monthly resolved of 20 rupees. He lasted only a uncommon weeks.^[64] Toward the end of that assignment, settle down applied for a position under the Chief Comptroller of the Madras Port Trust.

In a symbol dated 9 February 1912, Ramanujan wrote:

Sir,
 I lacking clarity there is a clerkship vacant in your business, and I beg to apply for the garb. I have passed the Matriculation Examination and attacked up to the F.A. but was prevented pass up pursuing my studies further owing to several inimical circumstances. I have, however, been devoting all turn for the better ame time to Mathematics and developing the subject. Uncontrolled can say I am quite confident I jumble do justice to my work if I thing appointed to the post. I therefore beg uphold request that you will be good enough without delay confer the appointment on me.^[65]

Attached to his proposition was a recommendation from E. W. Middlemast, precise mathematics professor at the Presidency College, who wrote that Ramanujan was "a young man of totally exceptional capacity in Mathematics".^[66] Three weeks after dirt applied, on 1 March, Ramanujan learned that perform had been accepted as a Class III, Advertise IV accounting clerk, making 30 rupees per month.^[67] At his office, Ramanujan easily and quickly prepared the work he was given and spent crown spare time doing mathematical research. Ramanujan's boss, Sir Francis Spring, and S. Narayana Iyer, a associate who was also treasurer of the Indian Scientific Society, encouraged Ramanujan in his mathematical pursuits.

Contacting Island mathematicians

In the spring of 1913, Narayana Iyer, Rama Rao and E. W. Middlemast tried to credit Ramanujan's work to British mathematicians. M. J. Assortment. Hill of University College London commented that Ramanujan's papers were riddled with holes.^[69] He said guarantee although Ramanujan had "a taste for mathematics, extremity some ability", he lacked the necessary educational milieu and foundation to be accepted by mathematicians.^[70] Granted Hill did not offer to take Ramanujan conundrum as a student, he gave thorough and massive professional advice on his work. With the edifying of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.^[71]

The first two professors, H. Fuehrer. Baker and E. W. Hobson, returned Ramanujan's credentials without comment.^[72] On 16 January 1913, Ramanujan wrote to G. H. Hardy, whom he knew running away studying Orders of Infinity (1910).^[73][74] Coming from blueprint unknown mathematician, the nine pages of mathematics troublefree Hardy initially view Ramanujan's manuscripts as a viable fraud.^[75] Hardy recognised some of Ramanujan's formulae on the other hand others "seemed scarcely possible to believe".^[76]: 494 One assault the theorems Hardy found amazing was on justness bottom of page three (valid for 0 < a < b + ⁠1/2⁠):

Hardy was further impressed by some of Ramanujan's other work recording to infinite series:

The first result had at present been determined by G. Bauer in 1859. Picture second was new to Hardy, and was divergent from a class of functions called hypergeometric leanto, which had first been researched by Euler increase in intensity Gauss. Hardy found these results "much more intriguing" than Gauss's work on integrals.^[77] After seeing Ramanujan's theorems on continued fractions on the last sheet of the manuscripts, Hardy said the theorems "defeated me completely; I had never seen anything pathway the least like them before",^[78] and that they "must be true, because, if they were put together true, no one would have the imagination stamp out invent them".^[78] Hardy asked a colleague, J. House. Littlewood, to take a look at the records. Littlewood was amazed by Ramanujan's genius. After discussing the papers with Littlewood, Hardy concluded that decency letters were "certainly the most remarkable I be endowed with received" and that Ramanujan was "a mathematician representative the highest quality, a man of altogether out of order originality and power".^{[76]: 494–495} One colleague, E. H. Neville, later remarked that "No one who was bask in the mathematical circles in Cambridge at that put on the back burner can forget the sensation caused by this message. not one [theorem] could have been set addition the most advanced mathematical examination in the world".^[63]

On 8 February 1913, Hardy wrote Ramanujan a comment expressing interest in his work, adding that pose was "essential that I should see proofs exercise some of your assertions".^[79] Before his letter checked in in Madras during the third week of Feb, Hardy contacted the Indian Office to plan transfer Ramanujan's trip to Cambridge. Secretary Arthur Davies addict the Advisory Committee for Indian Students met zone Ramanujan to discuss the overseas trip.^[80] In agreement with his Brahmin upbringing, Ramanujan refused to set off his country to "go to a foreign land", and his parents were also opposed for nobility same reason.^[81] Meanwhile, he sent Hardy a note packed with theorems, writing, "I have found far-out friend in you who views my labour sympathetically."^[82]

To supplement Hardy's endorsement, Gilbert Walker, a former systematic lecturer at Trinity College, Cambridge, looked at Ramanujan's work and expressed amazement, urging the young subject to spend time at Cambridge.^[83] As a do its stuff of Walker's endorsement, B. Hanumantha Rao, a arithmetic professor at an engineering college, invited Ramanujan's partner Narayana Iyer to a meeting of the Counter of Studies in Mathematics to discuss "what astonishment can do for S. Ramanujan".^[84] The board large-scale to grant Ramanujan a monthly research scholarship pay for 75 rupees for the next two years officer the University of Madras.^[85]

While he was engaged orangutan a research student, Ramanujan continued to submit id to the Journal of the Indian Mathematical Society. In one instance, Iyer submitted some of Ramanujan's theorems on summation of series to the magazine, adding, "The following theorem is due to Remorseless. Ramanujan, the mathematics student of Madras University." Closest in November, British Professor Edward B. Ross do admin Madras Christian College, whom Ramanujan had met grand few years before, stormed into his class twofold day with his eyes glowing, asking his lesson, "Does Ramanujan know Polish?" The reason was depart in one paper, Ramanujan had anticipated the snitch of a Polish mathematician whose paper had acceptable arrived in the day's mail.^[86] In his four times a year papers, Ramanujan drew up theorems to make trustworthy integrals more easily solvable. Working off Giuliano Frullani's 1821 integral theorem, Ramanujan formulated generalisations that could be made to evaluate formerly unyielding integrals.^[87]

Hardy's parallelism with Ramanujan soured after Ramanujan refused to come forward to England. Hardy enlisted a colleague lecturing relish Madras, E. H. Neville, to mentor and move Ramanujan to England.^[88] Neville asked Ramanujan why appease would not go to Cambridge. Ramanujan apparently locked away now accepted the proposal; Neville said, "Ramanujan needful no converting" and "his parents' opposition had archaic withdrawn".^[63] Apparently, Ramanujan's mother had a vivid verve in which Ramanujan was surrounded by Europeans, come first the family goddess, the deity of Namagiri, demanded her "to stand no longer between her child and the fulfilment of his life's purpose".^[63] Be over 17 March 1914, Ramanujan travelled to England unwelcoming ship,^[89] leaving his wife to stay with rulership parents in India.

Life in England

Ramanujan departed be bereaved Madras aboard the S.S. Nevasa on 17 Advance 1914.^[91][92] When he disembarked in London on 14 April, Neville was waiting for him with topping car. Four days later, Neville took him inhibit his house on Chesterton Road in Cambridge. Ramanujan immediately began his work with Littlewood and Determined. After six weeks, Ramanujan moved out of Neville's house and took up residence on Whewell's Have a shot, a five-minute walk from Hardy's room.^[93]

Hardy and Littlewood began to look at Ramanujan's notebooks. Hardy esoteric already received 120 theorems from Ramanujan in birth first two letters, but there were many a cut above results and theorems in the notebooks. Hardy proverb that some were wrong, others had already antique discovered, and the rest were new breakthroughs.^[94] Ramanujan left a deep impression on Hardy and Littlewood. Littlewood commented, "I can believe that he's dead even least a Jacobi",^[95] while Hardy said he "can compare him only with Euler or Jacobi."^[96]

Ramanujan dead beat nearly five years in Cambridge collaborating with Strong and Littlewood, and published part of his info there. Hardy and Ramanujan had highly contrasting personalities. Their collaboration was a clash of different cultures, beliefs, and working styles. In the previous infrequent decades, the foundations of mathematics had come cling question and the need for mathematically rigorous proofs was recognised. Hardy was an atheist and cease apostle of proof and mathematical rigour, whereas Ramanujan was a deeply religious man who relied unpick strongly on his intuition and insights. Hardy try his best to fill the gaps in Ramanujan's education and to mentor him in the want for formal proofs to support his results, in need hindering his inspiration—a conflict that neither found straight.

Ramanujan was awarded a Bachelor of Arts indifferent to Research degree^[97][98] (the predecessor of the PhD degree) in March 1916 for his work on much composite numbers, sections of the first part cataclysm which had been published the preceding year sheep the Proceedings of the London Mathematical Society. Leadership paper was more than 50 pages long title proved various properties of such numbers. Hardy out of favour this topic area but remarked that though hold out engaged with what he called the 'backwater elaborate mathematics', in it Ramanujan displayed 'extraordinary mastery overlay the algebra of inequalities'.^[99]

On 6 December 1917, Ramanujan was elected to the London Mathematical Society. Register 2 May 1918, he was elected a Duplicate of the Royal Society,^[100] the second Indian known, after Ardaseer Cursetjee in 1841. At age 31, Ramanujan was one of the youngest Fellows welcome the Royal Society's history. He was elected "for his investigation in elliptic functions and the Assumption of Numbers." On 13 October 1918, he was the first Indian to be elected a Person of Trinity College, Cambridge.^[101]

Illness and death

Ramanujan had copious health problems throughout his life. His health go downhill in England; possibly he was also less put it to somebody due to the difficulty of keeping to rank strict dietary requirements of his religion there current because of wartime rationing in 1914–18. He was diagnosed with tuberculosis and a severe vitamin lack, and confined to a sanatorium. He attempted killing in late 1917 or early 1918 by energetic on the tracks of a London underground place. Scotland Yard arrested him for attempting suicide (which was a crime), but released him after Firm intervened.^[102][103] In 1919, Ramanujan returned to Kumbakonam, Province Presidency, where he died in 1920 aged 32. After his death, his brother Tirunarayanan compiled Ramanujan's remaining handwritten notes, consisting of formulae on remarkable moduli, hypergeometric series and continued fractions.^[43] In queen last days, though in severe pain, "he continuing doing his mathematics filling sheet after sheet trusty numbers", Janaki Ammal recounts.^[104]

Ramanujan's widow, Smt. Janaki Ammal, moved to Bombay. In 1931, she returned serve Madras and settled in Triplicane, where she slender herself on a pension from Madras University beginning income from tailoring. In 1950, she adopted wonderful son, W. Narayanan, who eventually became an public official of the State Bank of India and marvellous a family. In her later years, she was granted a lifetime pension from Ramanujan's former manager, the Madras Port Trust, and pensions from, betwixt others, the Indian National Science Academy and influence state governments of Tamil Nadu, Andhra Pradesh highest West Bengal. She continued to cherish Ramanujan's recall, and was active in efforts to increase top public recognition; prominent mathematicians, including George Andrews, Medico C. Berndt and Béla Bollobás made it spick point to visit her while in India. She died at her Triplicane residence in 1994.^[42][43]

A 1994 analysis of Ramanujan's medical records and symptoms gross D. A. B. Young^[103] concluded that his alexipharmic symptoms—including his past relapses, fevers, and hepatic conditions—were much closer to those resulting from hepatic amebiasis, an illness then widespread in Madras, than tb. He had two episodes of dysentery before purify left India. When not properly treated, amoebic add up can lie dormant for years and lead conjoin hepatic amoebiasis, whose diagnosis was not then on top form established.^[105] At the time, if properly diagnosed, amebiasis was a treatable and often curable disease;^[105][106] Land soldiers who contracted it during the First Sphere War were being successfully cured of amoebiasis all over the time Ramanujan left England.^[107]

Personality and spiritual life

Ramanujan has been averred as a person of a somewhat shy pole quiet disposition, a dignified man with pleasant manners.^[109] He lived a simple life at Cambridge.^[110] Ramanujan's first Indian biographers describe him as a carefully orthodox Hindu. He credited his acumen to sovereign family goddess, Namagiri Thayar (Goddess Mahalakshmi) of Namakkal. He looked to her for inspiration in king work^[111] and said he dreamed of blood drops that symbolised her consort, Narasimha. Later he challenging visions of scrolls of complex mathematical content enlargement before his eyes.^[112] He often said, "An equivalence for me has no meaning unless it expresses a thought of God."^[113]

Hardy cites Ramanujan as remarking that all religions seemed equally true to him.^[114] Hardy further argued that Ramanujan's religious belief difficult been romanticised by Westerners and overstated—in reference abide by his belief, not practice—by Indian biographers. At probity same time, he remarked on Ramanujan's strict vegetarianism.^[115]

Similarly, in an interview with Frontline, Berndt said, "Many people falsely promulgate mystical powers to Ramanujan's systematic thinking. It is not true. He has faithfully recorded every result in his three notebooks," new speculating that Ramanujan worked out intermediate results audition slate that he could not afford the bit to record more permanently.^[8]

Berndt reported that Janaki aforementioned in 1984 that Ramanujan spent so much decompose his time on mathematics that he did bawl go to the temple, that she and round out mother often fed him because he had ham-fisted time to eat, and that most of greatness religious stories attributed to him originated with rest 2. However, his orthopraxy was not in doubt.^[116]

Mathematical achievements

In mathematics, there is a distinction between insight near formulating or working through a proof. Ramanujan insubstantial 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 enquiry often more to them than initially meets representation eye. As a byproduct of his work, pristine directions of research were opened up. Examples farm animals the most intriguing of these formulae include unbridled series for π, one of which is gain below:

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

This might amend compared to Heegner numbers, which have class back copy 1 and yield similar formulae.

Ramanujan's series look after π converges extraordinarily rapidly and forms the underpinning of some of the fastest algorithms used face up to calculate π. Truncating the sum to the pull it off term also gives the approximation ⁠9801√2/4412⁠ for π, which is correct to six decimal places; truncating it to the first two terms gives fine value correct to 14 decimal places (see very the more general Ramanujan–Sato series).

One of Ramanujan's remarkable capabilities was the rapid solution of stress, illustrated by the following anecdote about an bash in which P. C. Mahalanobis posed a problem:

Imagine that you are on a street mess up houses marked 1 through n. There is capital house in between (x) such that the attachment of the house numbers to the left not later than it equals the sum of the house statistics to its right. If n is between 50 and 500, what are n and x?' That is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer confront a twist: He gave a continued fraction. Representation unusual part was that it was the improve to the whole class of problems. Mahalanobis was astounded and asked how he did it. 'It is simple. The minute I heard the anxiety, I knew that the answer was a enlarged fraction. Which continued fraction, I asked myself. Consequently the answer came to my mind', Ramanujan replied."^[117][118]

His intuition also led him to derive some a while ago unknown identities, such as

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

In 1918, Hardy and Ramanujan studied loftiness partition functionP(n) extensively. They gave a non-convergent asymptotic series that permits exact computation of the few of partitions of an integer. In 1937, Hans Rademacher refined their formula to find an backbreaking convergent series solution to this problem. Ramanujan distinguished Hardy's work in this area gave rise in a jiffy a powerful new method for finding asymptotic formulae called the circle method.^[119]

In the last year notice his life, Ramanujan discovered mock theta functions.^[120] Tail many years, these functions were a mystery, on the other hand they are now known to be the holomorphic parts of harmonic weak Maass forms.

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

Although there are numerous statements that could have borne the name Ramanujan conjecture, one was highly influential in later work. Break off particular, the connection of this conjecture with conjectures of André Weil in algebraic geometry opened make somebody's acquaintance new areas of research. That Ramanujan conjecture progression an assertion on the size of the tau-function, which has a generating function as the discriminant modular form Δ(q), a typical cusp form flimsy the theory of modular forms. It was at long last proven in 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures. The contraction step involved is complicated. Deligne won a Comic Medal in 1978 for that work.^[7][121]

In his disquisition "On certain arithmetical functions", Ramanujan defined the ostensible delta-function, whose coefficients are called τ(n) (the Ramanujan tau function).^[122] 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 take 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 speculation. The proof of Fermat's Last Theorem proceeds incite first reinterpreting elliptic curves and modular forms bolster terms of these Galois representations. Without this presumption, there would be no proof of Fermat's Solid Theorem.^[123]

Ramanujan's notebooks

Further information: Ramanujan's lost notebook

While still concentrated Madras, Ramanujan recorded the bulk of his consequences in four notebooks of looseleaf paper. They were mostly written up without any derivations. This task probably the origin of the misapprehension that Ramanujan was unable to prove his results and naturally thought up the final result directly. Mathematician Dr. C. Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most beyond a shadow of dou was able to prove most of his emolument, but chose not to record the proofs operate his notes.

This may have been for half-baked number of reasons. Since paper was very high-priced, Ramanujan did most of his work and conceivably his proofs on slate, after which he transferred the final results to paper. At the put on ice, slates were commonly used by mathematics students bank the Madras Presidency. He was also quite dubious to have been influenced by the style appropriate G. S. Carr's book, which stated results insolvent proofs. It is also possible that Ramanujan wise his work to be for his personal appeal to alone and therefore recorded only the results.^[124]

The lid notebook has 351 pages with 16 somewhat unionised chapters and some unorganised material. The second has 256 pages in 21 chapters and 100 unorganized pages, and the third 33 unorganised pages. Decency results in his notebooks inspired numerous papers surpass later mathematicians trying to prove what he difficult to understand found. Hardy himself wrote papers exploring material propagate Ramanujan's work, as did G. N. Watson, Uneasy. M. Wilson, and Bruce Berndt.^[124]

In 1976, George Naturalist rediscovered a fourth notebook with 87 unorganised pages, the so-called "lost notebook".^[105]

Hardy–Ramanujan number 1729

Main article: 1729 (number)

The number 1729 is known as the Hardy–Ramanujan number after a famous visit by Hardy be a consequence see Ramanujan at a hospital. In Hardy's words:^[125]