Ramanujan, the mathematics student of Madras University. 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? Working off Giuliano Frullani's 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. Neville, to mentor and bring Ramanujan to England.
Ramanujan apparently had now accepted the proposal; Neville said, "Ramanujan needed no converting" and "his parents' opposition had been withdrawn".
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Ramanujan departed from Madras aboard the S. Nevasa on 17 March 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 had already received 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. 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.
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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 recognized. 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 Science degree by research this degree was later renamed PhD in March for his work on highly composite numbers , the first part of which was published as a paper in the Proceedings of the London Mathematical Society. The paper was more than 50 pages long and proved various properties of such numbers. Hardy remarked that it was one of the most unusual papers in mathematical research at that time and that Ramanujan showed extraordinary ingenuity in handling it.
At age 31 Ramanujan was one of the youngest Fellows in the history of the Royal Society. He was elected "for his investigation in Elliptic functions and the Theory of Numbers. Ramanujan was plagued by 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 — He was diagnosed with tuberculosis and a severe vitamin deficiency, and confined to a sanatorium.
In he returned to Kumbakonam , Madras Presidency , and in he died at the age of 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 she returned to Madras and settled in Triplicane , where she supported herself on a pension from Madras University and income from tailoring. In 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. She died at her Triplicane residence in A analysis of Ramanujan's medical records and symptoms by Dr. 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, dysentery can lie dormant for years and lead to hepatic amoebiasis, whose diagnosis was not then well established.
Ramanujan has been described as a person of a somewhat shy and quiet disposition, a dignified man with pleasant manners. He looked to her for inspiration in his work  : 36 and said he dreamed of blood drops that symbolised her consort, Narasimha. Afterward he would receive visions of scrolls of complex mathematical content unfolding before his eyes. Hardy cites Ramanujan as remarking that all religions seemed equally true to him.
At the same time, he remarked on Ramanujan's strict vegetarianism. 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. 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. This might be compared to Heegner numbers , which have class number 1 and yield similar formulae.
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. Mahalanobis posed a problem:. Imagine that you are on a street with houses marked 1 through n. There is a house in between x such that the sum of the house numbers to the left of it equals the sum of the house numbers to its right. If n is between 50 and , what are n and x?click
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Ramanujan thought about it and gave the answer with a twist: He gave a continued fraction. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind', Ramanujan replied.
His intuition also led him to derive some previously unknown identities , such as. In 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 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. Although there are numerous statements that could have borne the name Ramanujan conjecture, one was highly influential on later work.
It was finally proven in , as a consequence of Pierre Deligne 's proof of the Weil conjectures. The reduction step involved is complicated. Deligne won a Fields Medal in for that work. This congruence and others like it that Ramanujan proved inspired Jean-Pierre Serre Fields Medalist to conjecture that there is a theory of Galois representations that "explains" these congruences and more generally all modular forms.
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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.
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. This may have been for any number of reasons. Since paper was very expensive, Ramanujan would do most of his work and perhaps his proofs on slate , and then transfer just the results to paper.
Using a slate was common for mathematics students in the Madras Presidency at the time. He was also quite likely to have been influenced by the style of G.
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Carr 's book, which stated results without proofs. Finally, it is possible that Ramanujan considered his work to be for his personal interest alone and therefore recorded only the results. The first notebook has pages with 16 somewhat organised chapters and some unorganised material. The second has pages in 21 chapters and unorganised pages, and the third 33 unorganised pages.