Srinivasa Ramanujan

• 1. Introduction

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  Srinivasa Ramanujan was ill in a hospital at Putney when G. H. Hardy came to visit him. Hardy had ridden there in taxi cab number 1729. He remarked that the number seemed rather dull, and he hoped it was not an unfavorable omen. Ramanujan disagreed at once. He replied that it was "a very interesting number", and "the smallest number expressible as the sum of two cubes in two different ways." The idea has since grown into a family of curiosities known as taxicab numbers. Hardy's colleague J. E. Littlewood once said that every positive integer was one of Ramanujan's personal friends. Ramanujan was an Indian mathematician who left college in India without a degree, lived in poverty, and died young. Even so, researchers were still uncovering profound results hidden in his offhand comments as late as 2012. How did an unknown clerk in Madras win the attention of Cambridge? Why were so many established mathematicians unable to follow his work? And why do his notebooks still matter today?

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• 2. A Prodigy in Kumbakonam

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  Ramanujan was born on the 22nd of December 1887 in Erode, in what is now Tamil Nadu, into a Tamil Brahmin Iyengar family. His name means younger brother of Rama, a Hindu deity. His father, Kuppuswamy Srinivasa Iyengar, worked as a clerk in a sari shop. His mother, Komalatammal, kept the household and sang at a local temple. The family lived in a small traditional home on Sarangapani Sannidhi Street in Kumbakonam, and that home later became a museum.

  Grief came early to the household. When Ramanujan was a year and a half old, his mother gave birth to a son named Sadagopan, who died less than three months later. In December 1889 Ramanujan caught smallpox, though he recovered. Two more children, born in 1891 and 1894, each died before their first birthdays.

  On the 1st of October 1892 he was enrolled at the local school. The years that followed were unsettled. He and his mother went to her parents' house in Kanchipuram, near Madras, and returned to Kumbakonam after his maternal grandfather lost his job as a court official. After his paternal grandfather died, he was sent to his maternal grandparents, who were then living in Madras. He disliked school there and tried to avoid it, so the family enlisted a local constable to make sure he attended. Within six months he was back in Kumbakonam.

  Because his father was at work most of the day, his mother raised him. From her he learned the puranas, religious songs, temple pujas and the eating habits of Brahmin culture. In November 1897, just before turning 10, he passed his primary examinations in English, Tamil, geography and arithmetic with the best scores in the district.

  That year he entered Town Higher Secondary School and met formal mathematics for the first time. By the age of 11 he had exhausted what two college students lodging in his home could teach him. He then borrowed S. L. Loney's book on advanced trigonometry and mastered it by 13, discovering sophisticated theorems on his own. At 14 he was collecting merit certificates and helping the school assign its 1,200 students to its roughly 35 teachers. He completed mathematical exams in half the allotted time.

  In 1902 someone showed him how to solve cubic equations, and he later devised his own method for the quartic. In 1903 he tried to solve the quintic, not knowing that it cannot be solved with radicals.

  That same year, at 16, he obtained a library copy of G. S. Carr's A Synopsis of Elementary Results in Pure and Applied Mathematics from a friend. The book gathered 5,000 theorems, and Ramanujan reportedly studied it in detail. The next year he independently developed and investigated the Bernoulli numbers. He also calculated the Euler-Mascheroni constant to 15 decimal places. His classmates said they "rarely understood him" and "stood in respectful awe" of him.

  When he graduated in 1904, his headmaster, Krishnaswami Iyer, awarded him the K. Ranganatha Rao prize for mathematics. Iyer introduced him as an outstanding student who deserved scores higher than the maximum. A scholarship to Government Arts College in Kumbakonam came with the honor, though he would not be able to keep it.

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• 3. Poverty and Patrons

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  Mathematics so absorbed Ramanujan that he failed most of his other subjects at Government Arts College, and he lost his scholarship. In August 1905 he ran away from home, heading toward Visakhapatnam, and stayed about a month in Rajahmundry. He later enrolled at Pachaiyappa's College in Madras. There he passed in mathematics by attempting only the questions that appealed to him and leaving the rest unanswered. He did poorly in English, physiology and Sanskrit. He failed his Fellow of Arts exam in December 1906 and again a year later. Without that degree he left college and carried on researching alone, living in extreme poverty and often close to starvation.

  On the 14th of July 1909 he married Janaki, a girl his mother had selected a year earlier. She was ten years old, and such young matches were not unusual at the time. His father did not take part in the ceremony. Following custom, Janaki stayed at her maternal home for three years. In 1912 she and Ramanujan's mother joined him in Madras.

  Soon after the wedding, Ramanujan developed a hydrocele testis. A routine operation could treat it, but his family could not afford the cost. In January 1910 a doctor volunteered to perform the surgery for free. Ramanujan then went from door to door around Madras seeking a clerical job, staying at a friend's house. To earn money he tutored students at Presidency College who were preparing for their Fellow of Arts exam.

  In late 1910 he fell sick again and feared for his health. He told his friend R. Radakrishna Iyer to hand his notebooks to Professor Singaravelu Mudaliar of Pachaiyappa's College, or to the British professor Edward B. Ross of the Madras Christian College.

  That same year he met V. Ramaswamy Aiyer, a deputy collector who had founded the Indian Mathematical Society. Hoping for a job in Aiyer's revenue department, Ramanujan showed him the notebooks. Aiyer later recalled: "I was struck by the extraordinary mathematical results contained in the notebooks. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department."

  Aiyer sent Ramanujan with letters of introduction to mathematician friends in Madras. Some of them passed him on to R. Ramachandra Rao, the district collector for Nellore and secretary of the society. Rao was impressed but doubted the work was really Ramanujan's own. Ramanujan mentioned a Bombay mathematician, Professor Saldhana, who had written that he could not understand the work but did not think its author was a fraud. Ramanujan's friend C. V. Rajagopalachari also tried to ease Rao's doubts. Rao agreed to a second chance and listened as Ramanujan discussed elliptic integrals, hypergeometric series and his theory of divergent series. That conversation convinced him. When Rao asked what he wanted, Ramanujan answered that he needed work and financial support. Rao provided money for his research.

  With Aiyer's help, Ramanujan began publishing in the Journal of the Indian Mathematical Society. One of his first problems asked for the value of an infinitely nested radical. He waited through three issues, over six months, for a solution, and none arrived. In the end he supplied an incomplete solution himself. The answer was simply 3.

  His first formal paper for the Journal, in 1911, was on Bernoulli numbers. In 17 pages, "Some Properties of Bernoulli's Numbers" offered three proofs, two corollaries and three conjectures. The Journal's editor, M. T. Narayana Iyengar, observed that an ordinary reader, unaccustomed to such "intellectual gymnastics", could hardly follow him.

  In early 1912 Ramanujan took a temporary job in the Madras Accountant General's office at 20 rupees a month, and he lasted only a few weeks. In a letter dated the 9th of February 1912 he applied for a clerkship at the Madras Port Trust. He explained that he had studied up to the F.A., and that he had been devoting all his time to mathematics. Professor E. W. Middlemast of Presidency College backed the application, calling him "a young man of quite exceptional capacity in Mathematics". On the 1st of March he learned that he had been accepted as a Class III, Grade IV accounting clerk at 30 rupees a month. He finished his assigned work quickly and spent his spare time on research. His boss, Sir Francis Spring, and S. Narayana Iyer, a colleague who served as treasurer of the Indian Mathematical Society, encouraged his mathematical pursuits.

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• 4. Letters to Cambridge

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  Narayana Iyer, Ramachandra Rao and Middlemast tried in the spring of 1913 to put Ramanujan's work before British mathematicians. M. J. M. Hill of University College London found the papers riddled with holes. He granted that Ramanujan had "a taste for mathematics, and some ability", but said he lacked the educational foundation needed to be accepted. Hill still gave thorough and serious professional advice. The Cambridge professors H. F. Baker and E. W. Hobson returned the papers without comment.

  On the 16th of January 1913 Ramanujan wrote to G. H. Hardy, whom he knew through a work called Orders of Infinity from 1910. Nine pages of mathematics from an unknown correspondent first led Hardy to suspect a fraud. He recognised some of the formulae, but others, he said, "seemed scarcely possible to believe". One result had already been found by G. Bauer in 1859. Another was new to Hardy, and it came from hypergeometric series, first studied by Euler and Gauss. Hardy found these results much more intriguing than Gauss's work on integrals.

  The continued fractions on the last page affected him most. Hardy wrote that those theorems "defeated me completely; I had never seen anything in the least like them before". They "must be true, because, if they were not true, no one would have the imagination to invent them".

  Hardy asked his colleague J. E. Littlewood to look at the papers, and Littlewood was amazed. After talking them over, Hardy decided the letters were "certainly the most remarkable I have received". He judged Ramanujan "a mathematician of the highest quality, a man of altogether exceptional originality and power". Another colleague, E. H. Neville, later wrote that no one in Cambridge mathematical circles at the time could forget the sensation the letter caused. None of the theorems, he added, could have been set in the most advanced examination in the world.

  On the 8th of February 1913 Hardy replied that it was "essential that I should see proofs of some of your assertions." Before that letter reached Madras in the third week of February, he contacted the Indian Office about arranging a trip to Cambridge. Arthur Davies, secretary of the Advisory Committee for Indian Students, met with Ramanujan. In keeping with his Brahmin upbringing, Ramanujan refused to "go to a foreign land", and his parents opposed the idea for the same reason. He wrote to Hardy, "I have found a friend in you who views my labour sympathetically."

  Support grew at home. Gilbert Walker, a former mathematical lecturer at Trinity College, Cambridge, studied the work and urged Ramanujan to spend time there. Through B. Hanumantha Rao and the Board of Studies in Mathematics, the University of Madras granted him a research scholarship of 75 rupees a month for two years.

  In November, Edward B. Ross stormed into his class at Madras Christian College with his eyes glowing. He asked his students, "Does Ramanujan know Polish?" A paper by a Polish mathematician had arrived in that day's mail, and Ramanujan had anticipated its work. In his quarterly papers, Ramanujan also built on Giuliano Frullani's integral theorem of 1821. His generalisations helped evaluate integrals that had resisted solution.

  Still, the correspondence with Hardy soured over Ramanujan's refusal to travel. Hardy asked Neville, who was lecturing in Madras, to mentor Ramanujan and bring him to England. When Neville asked why he would not go, Ramanujan seemingly accepted the proposal. Neville said that "Ramanujan needed no converting" and that "his parents' opposition had been withdrawn." Ramanujan's mother had reportedly had a vivid dream in which her son was surrounded by Europeans. 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 the 17th of March 1914 Ramanujan sailed for England, leaving his wife with his parents in India.

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• 5. Hardy's Unlikely Partner

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  Neville was waiting with a car when Ramanujan stepped off the S.S. Nevasa in London on the 14th of April 1914. Four days later he took him to his house on Chesterton Road in Cambridge. Ramanujan began work with Littlewood and Hardy at once. After six weeks he moved to Whewell's Court, a five-minute walk from Hardy's room.

  Hardy already held 120 theorems from Ramanujan's first two letters, and the notebooks contained many more. Hardy and Littlewood found that some were wrong, others had been discovered before, and the rest were new breakthroughs. Littlewood said, "I can believe that he's at least a Jacobi". Hardy said he "can compare him only with Euler or Jacobi."

  The partnership lasted nearly five years, and it was a collision of temperaments. Mathematics was then questioning its own foundations, and rigorous proof was being demanded. Hardy was an atheist and an apostle of proof. Ramanujan was deeply religious and relied on intuition and insight. Hardy tried to fill the gaps in Ramanujan's education and to teach him why results need formal proofs, without hindering his inspiration. Neither found this easy.

  In March 1916 Ramanujan received a Bachelor of Arts by Research degree, the predecessor of the PhD. It was awarded for his work on highly composite numbers, and part of it had appeared the year before in the Proceedings of the London Mathematical Society. The paper ran past 50 pages. Hardy disliked the subject and called it a backwater of mathematics. Yet he said Ramanujan displayed "extraordinary mastery over the algebra of inequalities".

  Recognition followed quickly. On the 6th of December 1917 Ramanujan was elected to the London Mathematical Society. On the 2nd of May 1918 he became a Fellow of the Royal Society, the second Indian admitted, after Ardaseer Cursetjee in 1841. At 31 he was one of the youngest Fellows in the society's history. He was elected for his investigation in elliptic functions and the Theory of Numbers. On the 13th of October 1918 he became the first Indian elected a Fellow of Trinity College, Cambridge.

  Their most celebrated joint work came in 1918, on the partition function, which counts the ways an integer can be written as a sum of smaller integers. Hardy traced it to the 15 theorems he had examined in Ramanujan's first letter. He judged the first 13 correct and insightful, the 14th incorrect but insightful, and the 15th correct but misleading. The 14th, about the integer nearest to a certain coefficient, he called one of the most fruitful Ramanujan ever made. It "ended by leading us to all our joint work on partitions."

  The pair produced a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. Their work also gave rise to a powerful method for finding asymptotic formulae, called the circle method. In 1937 Hans Rademacher refined their formula into an exact convergent series.

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• 6. Illness and Return

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  Wartime rationing between 1914 and 1918 may have weakened Ramanujan, as may the difficulty of keeping the strict diet his religion required. His health, never robust, worsened in England. He was diagnosed with tuberculosis and a severe vitamin deficiency, and he was confined to a sanatorium. In late 1917 or early 1918 he attempted suicide by jumping onto the tracks of a London underground station. Scotland Yard arrested him, because attempting suicide was then a crime, but released him after Hardy intervened.

  In 1919 he returned to Kumbakonam, in the Madras Presidency, and he died there in 1920 at the age of 32. His last letters to Hardy, written in January 1920, show that he was still producing new theorems. His widow, Janaki Ammal, recalled that even in severe pain "he continued doing his mathematics filling sheet after sheet with numbers". His brother Tirunarayanan compiled the remaining handwritten notes, which covered singular moduli, hypergeometric series and continued fractions.

  In 1994 D. A. B. Young analysed Ramanujan's medical records and symptoms. The relapses, fevers and liver problems matched hepatic amoebiasis, an illness then widespread in Madras, far better than they matched tuberculosis. Ramanujan had suffered two episodes of dysentery before leaving India. Left untreated, amoebic dysentery can lie dormant for years before turning into the hepatic form. That form was not easily diagnosed at the time. If it had been recognised, it was often curable. British soldiers who caught amoebiasis in the First World War were being successfully cured around the time Ramanujan left England.

  After his death, Janaki Ammal moved to Bombay. She returned to Madras in 1931 and settled in Triplicane, supporting herself on a pension from Madras University and income from tailoring. In 1950 she adopted a son, W. Narayanan, who became an officer of the State Bank of India. In later years she received a lifetime pension from the Madras Port Trust, and further pensions from bodies including the Indian National Science Academy. The state governments of Tamil Nadu, Andhra Pradesh and West Bengal added their own. She stayed active in efforts to win Ramanujan wider recognition. Mathematicians including George Andrews, Bruce C. Berndt and Béla Bollobás made a point of visiting her while in India. She died at her Triplicane home in 1994.

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• 7. How He Thought

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  P. C. Mahalanobis once posed Ramanujan a puzzle about a street of houses numbered from 1 up to some number n. One house, he said, splits the street so that the numbers on its left add up to the same total as those on its right. If n lies between 50 and 500, which house is it, and what is n? Ramanujan answered with a continued fraction, which turned out to solve the whole class of such problems and not just the one asked. Mahalanobis was astounded and asked how he did it. Ramanujan replied, "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."

  Hardy said Ramanujan's results were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account." In his 1920 obituary for Nature, Hardy drew a startling portrait. Here was a man who could work out modular equations and theorems to orders unheard of. His mastery of continued fractions was beyond that of any mathematician in the world. Yet he had never heard of a doubly periodic function or of Cauchy's theorem, and had only the vaguest idea what a function of a complex variable was.

  Hardy speculated about what might have happened had Ramanujan met modern methods at sixteen instead of twenty-six. He thought it not extravagant to suppose he might have become the greatest mathematician of his time. He also thought Ramanujan worked in a nineteenth-century style, where reaching correct formulas mattered more than building systematic theories. In analytic number theory, where rigorous proof matters more than imagination, much of what Ramanujan imagined was wrong. Hardy called his "one great failure" that he knew nothing at all about the theory of analytic functions. Still, Hardy wrote that, "Playing the game of which he knew the rules, he could give any mathematician in the world fifteen."

  Littlewood reportedly found it very hard to bring Ramanujan up to date with European mathematics. Each new point he mentioned set off original ideas in Ramanujan's mind, and the lesson could not continue.

  Ramanujan credited his acumen to his family goddess, Namagiri Thayar of Namakkal. He said he dreamed of blood drops that symbolised her consort, Narasimha. Later he had visions of scrolls of complex mathematics unfolding before his eyes. He often said, "An equation for me has no meaning unless it expresses a thought of God." Hardy cited Ramanujan as remarking that all religions seemed equally true to him. Hardy argued that Westerners had romanticised Ramanujan's belief, and that Indian biographers had overstated it. Berndt told the magazine Frontline that many people falsely attribute mystical powers to Ramanujan's thinking. Berndt said Ramanujan had meticulously recorded every result in his notebooks. Berndt also reported that in 1984 Janaki said he spent so much time on mathematics that he did not go to the temple. She said that she and her mother often fed him because he had no time to eat, and that most of the religious stories attributed to him came from others.

  Berndt passed on a scale of pure talent from 0 to 100 that Paul Erdős attributed to Hardy. Hardy gave himself 25 and Littlewood 30. He gave David Hilbert 80, and Ramanujan 100.

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• 8. Notebooks and Legacy

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  The first of Ramanujan's notebooks has 351 pages, with 16 somewhat organised chapters and some unorganised material. The second has 256 pages in 21 chapters, plus 100 unorganised pages. The third has 33 unorganised pages. Most results were written without derivations, which probably gave rise to the false idea that Ramanujan could not prove his results and simply thought up the answers. Berndt says he most certainly could prove most of them but chose not to record the proofs. Paper was expensive, and slates were commonly used by mathematics students in the Madras Presidency, so he probably worked on slate and copied only the final results. He may also have been influenced by Carr's book, which stated results without proofs, or have seen the work as for his own interest alone.

  Those notebooks have inspired numerous papers by later mathematicians trying to prove what he found. Hardy wrote some himself, as did G. N. Watson, B. M. Wilson and Berndt. In 1976 George Andrews rediscovered a fourth notebook of 87 unorganised pages, the so-called lost notebook. It held discoveries from the last year of Ramanujan's life, and it caused great excitement among mathematicians. Andrews and Berndt later published it with commentary in a series of five volumes. The Ramanujan Journal was founded to publish work in all areas of mathematics he influenced, and most of his thousands of results have been proven correct.

  One famous example is his infinite series for pi, which converges extraordinarily rapidly and forms the basis of some of the fastest algorithms for calculating pi. Truncating the series to its first term gives an approximation correct to six decimal places. Truncating it to the first two terms gives a value correct to 14.

  A conjecture about the size of the Ramanujan tau function, tied to a modular form called the discriminant, shaped decades of later work. It connected with the Weil conjectures of André Weil, and it was finally proven in 1973 as a consequence of Pierre Deligne's proof of those conjectures. Deligne won a Fields Medal in 1978. Congruences that Ramanujan proved for the tau function inspired Jean-Pierre Serre to conjecture a theory of Galois representations that would explain them. Deligne proved that conjecture too. The proof of Fermat's Last Theorem begins by reinterpreting elliptic curves and modular forms through these representations. Without this theory, there would be no such proof. In the last year of his life, Ramanujan also discovered mock theta functions. They remained a mystery for many years, and are now known as the holomorphic parts of harmonic weak Maass forms.

  During a May 2011 lecture at IIT Madras, Berndt said that nearly all of Ramanujan's conjectures had been proven over the previous 40 years. He said Ramanujan's work was pervading many areas of modern mathematics and physics.

  In 2011, on the 125th anniversary of his birth, the Indian government declared that the 22nd of December would be celebrated each year as National Mathematics Day. Prime Minister Manmohan Singh also declared 2012 National Mathematics Year. Tamil Nadu marks the same date as State IT Day, and India issued stamps bearing his portrait in 1962, 2011-2012 and 2016. Dev Patel played Ramanujan in the 2015 film The Man Who Knew Infinity, which was based on Kanigel's book of the same name.

  The SASTRA Ramanujan Prize of US$10,000 is awarded each year for work in areas he influenced. It goes to a mathematician no older than 32, the age at which Ramanujan died.

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