Srinivasa Ramanujan

Srinivasa Ramanujan was born on December 22, 1887, in Erode, a town in the Madras Presidency of British India, into a Tamil Brahmin family of modest means ^[1]. His father worked as a clerk in a cloth merchant's shop in Kumbakonam, the temple town where the family lived for most of Ramanujan's childhood, and his mother sang at a local temple and managed the household ^[2].

The boy's talent surfaced early. At Town Higher Secondary School in Kumbakonam he outpaced his classmates in arithmetic and, by his early teens, was working through advanced trigonometry on his own and devising theorems that his teachers could not check ^[1]. Around the age of 16 he borrowed a copy of G. S. Carr's A Synopsis of Elementary Results in Pure and Applied Mathematics, a compressed reference book listing roughly 5,000 theorems with little or no proof. The book became his template: Ramanujan absorbed its contents, then began generating results of his own in the same terse style, recording them in notebooks that would later become famous ^[3].

The obsession carried a cost. Ramanujan won a scholarship to Government Arts College in Kumbakonam in 1904, but he neglected every subject except mathematics, failed his other examinations, and lost the scholarship. A second attempt at Pachaiyappa's College in Madras ended the same way ^[2]. By his early twenties he had no degree, no job, and a growing stack of notebooks filled with mathematics that almost nobody around him could read.

Who was Srinivasa Ramanujan in these years? To his neighbors, a poor young man without prospects. In 1909 he entered an arranged marriage with Janaki Ammal, then a young girl in keeping with the customs of the time, and the need to support a household pushed him to look seriously for work ^[2]. He tutored students, showed his notebooks to anyone who might help, and eventually found patrons among Indian mathematics enthusiasts, including Ramaswamy Aiyer, founder of the Indian Mathematical Society, and R. Ramachandra Rao, a district collector who supported him for a time ^[1].

In 1912 Ramanujan secured a clerical post as an accountant in the Madras Port Trust. The job paid little but left him time for mathematics, and his supervisors, notably the office manager S. Narayana Iyer and the Port Trust chairman Sir Francis Spring, recognized that their clerk was doing something unusual and encouraged him ^[3]. His first published paper, on properties of Bernoulli numbers, appeared in the Journal of the Indian Mathematical Society in 1911 ^[1].

The decisive step came in January 1913, when Ramanujan wrote to G. H. Hardy, a leading analyst at Trinity College, Cambridge. The letter contained pages of theorems: some known, some slightly wrong, and some so strange that Hardy later said they had to be true because no one could have had the imagination to invent them ^[4]. Hardy consulted his collaborator J. E. Littlewood, and the two concluded they were reading the work of a mathematician of exceptional power. Hardy arranged support, and after initial religious objections to crossing the sea were resolved, Ramanujan sailed for England in March 1914 ^[2].

The five years Ramanujan spent in Cambridge produced most of the results usually listed among Srinivasa Ramanujan achievements. Working with Hardy, he developed the circle method to derive an astonishingly accurate asymptotic formula for the partition function p(n), which counts the ways a whole number can be split into sums of positive integers ^[4]. The Hardy-Ramanujan collaboration also produced a foundational result on the normal number of prime factors of an integer, a theorem that helped launch probabilistic number theory ^[5].

Ramanujan's independent work ranged across infinite series, continued fractions, elliptic functions, and what he called mock theta functions. His series for 1/pi, published in 1914, converge so rapidly that variants of them powered computer calculations of pi decades later ^[5]. He introduced the tau function and made conjectures about its properties that occupied number theorists for most of the twentieth century; one of them, the Ramanujan conjecture, was finally settled by Pierre Deligne's work in the 1970s, which earned a Fields Medal ^[4].

Recognition followed. Ramanujan received a Bachelor of Science by research from Cambridge in 1916 for a dissertation on highly composite numbers. In 1918 he was elected a Fellow of the Royal Society, one of the youngest fellows in the society's history and among the first Indians so honored, and later that year Trinity College elected him to a fellowship, the first Indian to receive one ^[1]. Among Srinivasa Ramanujan facts, one anecdote is repeated more than any other: when Hardy visited him in hospital and remarked that his taxi's number, 1729, seemed dull, Ramanujan replied at once that it was the smallest number expressible as the sum of two cubes in two different ways. Such numbers are now called taxicab numbers ^[4].

Ramanujan was a devout Hindu throughout his life, a strict vegetarian, and a devotee of his family deity, the goddess Namagiri of Namakkal. He told colleagues that his theorems came to him in visions granted by the goddess, and he is often quoted as saying that an equation had no meaning for him unless it expressed a thought of God ^[2]. Hardy, a committed atheist, was skeptical of the framing but never doubted the mathematics.

His marriage to Janaki Ammal was typical of its era: arranged when she was a child, with the couple living together only briefly before his departure for England. Janaki remained in India during his Cambridge years, and relations between her and Ramanujan's strong-willed mother were strained. After his death Janaki lived quietly in Madras for decades, surviving until 1994 and receiving belated recognition and pensions in her later years ^[3].

Life in wartime England was hard on him. Vegetarian food was scarce during the First World War, he cooked for himself, worked through the night, and grew increasingly isolated and unwell. In 1917 he fell seriously ill, spent long stretches in sanatoriums, and at a low point in early 1918 attempted to take his own life on the tracks of the London Underground; he survived and the matter was quietly handled with Hardy's intervention ^[2].

Ramanujan's health never recovered in England. His illness was diagnosed at the time as tuberculosis complicated by severe vitamin deficiency, though a later analysis by the physician D. A. B. Young argued that the pattern of symptoms better fits hepatic amoebiasis, a parasitic liver infection he likely contracted in India and that was treatable even then ^[6]. Whatever the cause, he continued to work from his sickbed, producing results between hospital stays.

He returned to India in 1919, arriving in Madras to a reputation utterly transformed: the failed college student was now a Fellow of the Royal Society. But the homecoming brought no cure. Cared for by Janaki in Madras and then at Kumbakonam, he kept doing mathematics almost to the end, and in January 1920 wrote a final letter to Hardy describing a new class of functions he called mock theta functions ^[4].

Ramanujan died on April 26, 1920, at the age of 32 ^[1]. The papers from his last year, rediscovered by George Andrews in the Trinity College library in 1976 and known as the lost notebook, contained about 600 further results. Mathematicians spent the following decades proving and interpreting them, and the mock theta functions found a rigorous framework only in the 2000s through work on harmonic Maass forms ^[5].

Few mathematicians of any era have left so much unfinished business for their successors. Ramanujan's three earlier notebooks, containing several thousand results mostly stated without proof, were systematically edited and proved over more than two decades by Bruce Berndt, whose five-volume series Ramanujan's Notebooks appeared between 1985 and 1998 ^[5]. Berndt later collaborated with George Andrews on a further multi-volume treatment of the lost notebook. The Ramanujan Journal, founded in 1997, publishes research in the areas his work touched.

His ideas turned out to reach far beyond pure number theory. Ramanujan graphs are used in computer science and network design, his partition congruences connect to the theory of modular forms, and mock theta functions have appeared in the study of black hole entropy in string theory ^[5]. In India he is a national icon: his birthday, December 22, has been observed as National Mathematics Day since 2012, and the SASTRA Ramanujan Prize honors young mathematicians working in his fields ^[1].

The life has also entered popular culture. Robert Kanigel's 1991 biography The Man Who Knew Infinity brought the story to a wide readership and was adapted into a 2016 film starring Dev Patel as Ramanujan and Jeremy Irons as Hardy ^[7]. Hardy himself supplied the most quoted verdict: asked about his own greatest contribution to mathematics, he said it was the discovery of Ramanujan, and he described their association as the one romantic incident of his life ^[4].