how Ramanujan got to Cambridge

The Mechanism

Srinivasa Ramanujan was born in 1887 in Erode, Madras Presidency, in a Tamil Brahmin family too poor to keep him in school. He borrowed a copy of G. S. Carr's *A Synopsis of Elementary Results in Pure and Applied Mathematics* — a compressed 1886 handbook of about six thousand theorems, printed largely without proofs — at fifteen and worked through it obsessively over four years, then invented the rest of what he needed for himself. He failed out of college twice for neglecting every subject that was not mathematics. By 1912 he was 24, married, and desperate for a livelihood, and had a job as a Class III Grade IV accounts clerk at the Madras Port Trust, making thirty rupees a month, on the recommendation of Sir Francis Spring, the harbour engineer, who had been asked by mathematically-inclined intermediaries to give the young man a stable income while he continued to fill his notebooks with mathematics in the evenings. His superior at the Port Trust, S. Narayana Iyer, was himself a mathematician who had published a paper on prime numbers. Iyer and the chairman Spring encouraged Ramanujan to send his best results to a mathematician in England. He wrote first to two Cambridge dons — H. F. Baker and E. W. Hobson — and got no reply from either. On 16 January 1913 he tried a third: Godfrey Harold Hardy, 35, Fellow of Trinity College Cambridge, Britain's leading pure mathematician. The letter began: "I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of only £20 per annum. I am now about 23 years of age. I have had no University education … after leaving school I have been employing the spare time at my disposal to work at mathematics. I have not trodden through the conventional regular course …" What followed was nine pages of densely written mathematics: about 120 formulae, including new results on the gamma function, on divergent series, on the distribution of prime numbers, and on continued fractions — with almost no proofs. Hardy received the letter at Trinity around 31 January 1913. His first reaction was that it was a hoax; Cambridge got its share of correspondence from cranks who thought they had trisected the angle or squared the circle. He put the letter aside and went to lunch. Over dinner that night in Hall with his collaborator John Edensor Littlewood, he brought the pages back out. They spent the evening in Littlewood's rooms working through the formulae. Some Hardy already knew, but they were correctly restated; some he could see how to prove; some he was unable to prove but suspected might be true; and some were results he had never seen anyone state before. Some of these last, he later wrote, were results "which defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class. They must be true because, if they were not true, no one would have had the imagination to invent them." Hardy wrote back to Madras on 8 February 1913 — the reply Ramanujan had been waiting for since he was fifteen. In March 1914 Ramanujan sailed for England. He worked with Hardy at Trinity College Cambridge for five years, coauthoring the Hardy-Ramanujan asymptotic formula for the partition function *p(n)* (the number of ways to write *n* as a sum of positive integers), the "circle method" of analytic number theory, and Ramanujan's mock theta functions. In 1918 he was elected a Fellow of the Royal Society at 30, one of the youngest fellows in the Society's history and the second Indian ever elected. Tuberculosis and malnutrition — worsened by his religious vegetarianism during wartime Cambridge food shortages — sent him home to Madras in 1919. He died there at 32 in April 1920, filling his last three notebooks with new identities on his deathbed; the "lost notebook" of that final year, rediscovered by George Andrews in the Trinity College library in 1976, is still being mined for open problems in analytic number theory a century later. Hardy said until his death that his greatest contribution to mathematics was Ramanujan.