A Madras Clerk's Letter of 16 January 1913 Held Formulas That Defeated G. H. Hardy, and Ramanujan's Last Letter of 1920 Waited 82 Years for a Theory to Explain It

Srinivasa Ramanujan was a clerk in the accounts department of the Madras Port Trust when he wrote to G. H. Hardy of Trinity College, Cambridge, on 16 January 1913. He gave his salary as £20 a year and his age as about 23. By the birth date Hardy printed later, 22 December 1887, he was 25.

Seven years later he was dead at 32. In between he had been elected to the Royal Society and to a Trinity fellowship, had written with Hardy a formula for the number of partitions of a whole number, and had sent one last letter whose functions had no proper definition until 2002. Most of what follows comes from Hardy’s own writings, the Royal Society and Trinity College. Where sources disagree, the text says so.

The envelope from Madras

Ramanujan’s covering letter is printed in full in Hardy’s 1921 obituary notice, reprinted in the Collected Papers of 1927. It begins with his credentials: “I have had no University education but I have undergone the ordinary school course.” Further down he writes “I am striking out a new path for myself”, and he ends with a request: “Being poor, if you are convinced that there is anything of value I would like to have my theorems published.”

The path had started in 1903 at the Government College in Kumbakonam, when a friend borrowed Carr’s Synopsis of Pure Mathematics from the library for him. P. V. Seshu Aiyar and R. Ramachandra Rao, the Madras friends who wrote his Indian biographical notice, say “it was this book that awakened his genius”. Hardy adds that Ramanujan failed the First Examination in Arts, a university exam, and never tried again.

How much he sent depends on who is counting. The Madras notice says the enclosed papers held a hundred or more theorems. Hardy later wrote of about 120 theorems in Ramanujan’s early letters taken together. C. P. Snow, retelling the story in his foreword to Hardy’s A Mathematician’s Apology, put the first packet at nine pages of formulas.

Hardy was not his first try. MacTutor, the mathematical biography archive at the University of St Andrews, says Ramanujan had also written to E. W. Hobson and H. F. Baker, and neither replied.

The formulas that beat Hardy

Hardy described his first reading in a lecture at Harvard in 1936, printed in his 1940 book. Some formulas he recognised: one was a result of Laplace first proved properly by Jacobi, and another appeared in a paper by L. J. Rogers in 1907. The integrals he could prove himself, “though with a good deal more trouble than I had expected”.

Three formulas about continued fractions, which are fractions whose denominators contain further fractions without end, were different. Hardy said they “defeated me completely”. He went on: “They must be true because, if they were not true, no one would have had the imagination to invent them.” Don Zagier, writing in 2007, says Hardy spent the night with J. E. Littlewood convincing himself that the letter came from a genius and not a fraud. Hardy answered on 8 February 1913, MacTutor reports, saying he was “exceedingly interested” and asking for proofs.

Some of the claims were wrong. Hardy wrote in 1921 that Ramanujan’s Indian work on prime numbers “was definitely wrong”, because his methods ignored what complex numbers do to such sums. A second false statement, about the size of the coefficients in a power series, Hardy called one of the most fruitful Ramanujan ever made, “since it ended by leading us to all our joint work on partitions”. Of the fifteen formulas Hardy printed in 1940, he says all the rest had been verified by somebody since, with Rogers and G. N. Watson proving the three that defeated him.

A year to get him on the boat

Madras moved faster than Cambridge. Sir Francis Spring, chairman of the Port Trust, showed Ramanujan’s work to G. T. Walker, then head of the Meteorological Department in India and a former Trinity fellow. Walker’s letter of 26 February 1913 to the University of Madras led to a research scholarship of Rs 75 a month for two years. The Madras notice says Ramanujan left the Port Trust office on 1 May 1913 and “remained for the rest of his life a professional mathematician”.

Hardy wanted him in Cambridge. According to the Madras notice, Ramanujan first declined because of caste scruples. Hardy kept writing, and in early 1914 he asked E. H. Neville, a Trinity fellow lecturing at Madras, to press the case. Neville’s memorandum of 28 January 1914 called the discovery of Ramanujan’s genius “the most interesting event of our time in the mathematical world”. The notice adds that the last obstacle was his mother, who agreed after a dream in which, she said, the goddess Namagiri told her not to stand in his way. That detail is the Madras friends’ account and rests on their word.

Ramanujan sailed on 17 March 1914 and reached Cambridge in April, in Neville’s company, Hardy says. He had £250 a year from Madras, of which £50 went to his family, and a £60 exhibition from Trinity. Hardy wrote that he kept the religious observances of his caste in England “with a severity most unusual” among Indians there.

Teaching a man who had not heard of Cauchy’s theorem

Hardy faced a problem he described plainly in 1921. Here was a man who had found the functional equation of the zeta function by himself, and whose handling of continued fractions, on the formal side, Hardy thought beyond any mathematician’s in the world, yet “had never heard of a doubly periodic function or of Cauchy’s theorem”, a basic tool for complex numbers. Hardy feared that too much insistence on proof would break Ramanujan’s confidence. He tried anyway, and wrote that “obviously I learnt from him much more than he learnt from me”.

The First World War took Littlewood away, so Hardy stayed the teacher. By K. Srinivasa Rao’s count Ramanujan published 21 papers in England, five of them with Hardy. Hardy’s own list adds two joint notes. One, read to the London Mathematical Society on 14 December 1916, proved that almost every number n has about log log n prime factors. In the Apology Hardy wrote that “my association with them”, meaning Littlewood and Ramanujan, “was the decisive event of my life”.

Ramanujan’s longest paper, from 1915, was his alone. Highly composite numbers are those with more divisors than any smaller number. Twelve has six divisors and sixty has twelve, and both qualify. Rao says the paper runs to 62 pages and that Ramanujan received a Cambridge BA by research in March 1916 on the strength of it. Hardy thought it work “in a backwater of mathematics” but admired the handling of inequalities. His 1914 paper on modular equations and approximations to π gave algebraic approximations, two of which Hardy quotes as correct to nine and eight decimal places. The Kerala school had found infinite series for π centuries earlier.

A partition formula that missed by four thousandths

A partition of a whole number is a way of writing it as a sum of smaller whole numbers, ignoring order. The number 4 has five: 4, 3+1, 2+2, 2+1+1 and 1+1+1+1. Counting them gets difficult quickly. Major P. A. MacMahon had tabulated the count, written p(n), for every n up to 200.

Hardy and Ramanujan announced their formula in a note to the French Comptes Rendus dated 2 January 1917 and printed the full version in the Proceedings of the London Mathematical Society in 1918. Their paper compared the first six terms of the series with MacMahon’s exact values. For n = 100 the six terms gave 190,569,291.996 against an exact p(100) of 190,569,292. For n = 200 they gave 3,972,999,029,388.004 against 3,972,999,029,388. The error in both was four thousandths.

The authors admitted a gap. They had never been able to prove, they wrote, that the series sums to p(n) exactly, “nor even that it is convergent”. In 1937 D. H. Lehmer showed that their infinite series diverges, and Hans Rademacher published a changed series that converges to p(n) exactly, in the Proceedings of the National Academy of Sciences. Hardy called Rademacher’s change “very fortunate”. Studying MacMahon’s table, Ramanujan noticed that p(5n+4) appeared always to be a multiple of 5, p(7n+5) of 7 and p(11n+6) of 11.

Fellow of the Royal Society at 30, and ill

The Royal Society certificate for Ramanujan was delivered on 18 December 1917, according to Rao. Hardy proposed him and MacMahon seconded. Among the signatories were Hobson and Baker, the two mathematicians who, by MacTutor’s account, had not answered his letters in 1913.

The date of election differs by source. The Madras notice and Rao give 28 February 1918, while the Royal Society’s own account says he was made a Fellow on 2 May 1918. The Madras notice also calls him the first Indian elected. The Society itself names Ardaseer Cursetjee, elected on 27 May 1841, as its first Indian Fellow. The Nobel Foundation says the physicist CV Raman was elected in 1924.

Trinity elected him to a six-year prize fellowship in October 1918, worth about £250 a year with no duties, the Madras notice says. Announcing it to the University of Madras, Hardy wrote that Ramanujan “will return to India with a scientific standing and reputation such as no Indian has enjoyed before”.

By then he had been ill for a year. From the spring of 1917 he was in a nursing home at Cambridge and then in sanatoria at Wells, Matlock and London, and Hardy says he improved noticeably only in autumn 1918. What was wrong is unsettled. Doctors considered tuberculosis among other conditions, and the Madras notice speaks of a tubercular tendency. In 1994 D. A. B. Young argued in Current Science for hepatic amoebiasis, a parasitic infection of the liver, as Rao summarises it.

The taxi at Putney

In his obituary notice and again in the 1940 lectures, Hardy tells one story. He visited Ramanujan, who was ill at Putney in south-west London. Ramanujan’s own letter of 11 January 1919 to the University of Madras is headed 2 Colinette Road, Putney. Hardy wrote that he “had ridden in taxi-cab No. 1729” and remarked that the number seemed to him “rather a dull one”. Ramanujan replied: “it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways.”

The two ways are 1³ + 12³ and 9³ + 10³, and both come to 1729. Hardy then asked about the same problem for fourth powers. Ramanujan thought for a moment, said he knew no obvious example and supposed the first such number must be very large. Hardy gives no date for the visit, and the story comes from him alone.

Retellings have grown. Snow’s foreword gives Hardy an abrupt opening line and Ramanujan a doubled “No, Hardy!”, as John Baez quotes it. Ken Ono and Sarah Trebat-Leder point out that Ramanujan did not need to improvise. His second notebook, written before he left India, already lists 1729 as a sum of two cubes in two ways. Writing a number like 1729 at all relies on the place-value counting and zero described in India gave the world zero.

Home to Madras, and the last letter

Ramanujan left England on 27 February 1919 and reached Bombay on 27 March, the Madras notice says. He was thin and pale. He spent time in Madras, in Kodumudi and in Kumbakonam before going back to Madras for treatment in January 1920. He died on 26 April 1920. Hardy’s notice puts his death at Kumbakonam, while Seshu Aiyar and Ramachandra Rao place it at Chetput, a suburb of Madras.

The last letter is dated 12 January 1920, from the University of Madras. Ramanujan told Hardy he had found functions he called “mock” theta functions, and listed 17 examples without proofs or a definition. Zagier counts four of order 3, ten of order 5 and three of order 7. Hardy says the letter reached him in February 1920 and that he was “quite unprepared for the news of his death”.

Watson proved Ramanujan’s statements for the order 3 functions in a lecture of 1935. No one had a definition of what the functions were. In spring 1976 George Andrews opened a box of papers in the Wren Library at Trinity, sent there from Watson’s effects, and found Ramanujan’s own pages on the mock theta functions, as Robert Schneider’s interviews relate. Sander Zwegers’s 2002 doctoral thesis at Utrecht, supervised by Zagier and R. W. Bruggeman, showed that Ramanujan’s examples fall into three families. Adding a specific correction term to each gives a function with the symmetries of a modular form, a class for which Ramanujan’s word was “theta function”.

Trinity College says Ramanujan compiled nearly 3,900 results in his notebooks, mostly identities and equations, and that most have proved correct. The Government of India declared his birthday, 22 December, National Mathematics Day in December 2011.