The Man Who Knew Infinity

by Robert Kanigel

  India was a place of spiritual values triumphant? Ramanujan, at least up until the very end of his life and perhaps even then, never forsook the Indian gods. He invoked the name of the goddess Namagiri. He was steeped in, and accepted, most of the values, beliefs, and ways of life of South India.

  The Indian psychologist Ashis Nandy has made a similar point in his book Alternative Sciences, contrasting Ramanujan with the Indian physicist and plant physiologist J. C. Bose. Bose, he wrote, was ever troubled by the split within him between East and West, while Ramanujan, the more autonomous of the two, was not. “In his nationalism, as much as in his Westernized modernity, [Bose] was far more deeply bound up with the West, both in admiration and in hatred.” Ramanujan, on the other hand, was who he was—a South Indian Brahmin through and through. Less than many other Indians was Ramanujan’s head turned by the West and its ways.

  Not through anything he professed but in the very model of his life and his achievements, Ramanujan lived on in the soul of India. “The British thought Indians were inferior, and Ramanujan showed otherwise,” says P. K. Srinivasan, a Madrasi mathematics teacher who has compiled a book of reminiscences about Ramanujan. “He boosted our morale.” In E. H. Neville’s view it was the particular playing field on which Ramanujan triumphed—mathematics, in all its austere purity—that magnified his impact. In 1941, Neville wrote out a radio lecture on Ramanujan. His talk was broadcast but, whether due to constraints of time or politics, he did not deliver all he had prepared. Among comments left unaired were these:

  Ramanujan’s career, just because he was a mathematician, is of unique importance in the development of relations between India and England. India has produced great scientists, but Bose and Raman were educated outside India, and no one can say how much of their inspiration was derived from the great laboratories in which their formative years were spent and from the famous men who taught them. India has produced great poets and philosophers, but there is a subtle tinge of patronage in all commendation of alien literature. Only in mathematics are the standards unassailable, and therefore of all Indians, Ramanujan was the first whom the English knew to be innately the equal of their greatest men. The mortal blow to the assumption, so prevalent in the western world, that white is intrinsically superior to black, the offensive assumption that has survived countless humanitarian arguments and political appeals and poisoned countless approaches to collaboration between England and India, was struck by the hand of Srinivasa Ramanujan.

  In London, at a meeting of the Mathematical Society on June 10, 1920, according to its minutes, “The President referred to the loss that the Society has suffered in the death of Mr. S. Ramanujan and Major MacMahon spoke on the subject of Mr. Ramanujan’s mathematical work.”

  Hardy, after so recently getting the seemingly upbeat letter on mock theta functions, was shocked to learn that Ramanujan had died. He wrote a brief obituary for Nature, to which Neville added more information some months later, then a much longer one for the Proceedings of the London Mathematical Society.

  In America, the American Mathematical Monthly, expanding the section it normally reserved for dry accounts of papers appearing in foreign journals, gave ample play to the romantic story of Ramanujan’s discovery, even down to his physical appearance: “In conversation he became animated, and gesticulated vividly with his slender fingers.”

  Ramanujan’s papers had somehow wound up with the University of Madras; Janaki later charged Seshu Iyer with spiriting them away during the funeral itself. A few years later, on August 30, 1923, Dewsbury sent all but the original notebooks to Hardy.

  In 1921, Hardy brought out the last of Ramanujan’s papers, and the same year, he began to take up Ramanujan’s work in papers of his own. The early 1920s saw others do so as well. In 1922, Mordell came out with “Note on Certain Modular Relations Considered by Messrs. Ramanujan, Darling and Rogers.” The following year, B. M. Wilson’s “Proofs of Some Formulae Enunciated by Ramanujan” was published in the Proceedings of the London Mathematical Society.

  Soon after learning of Ramanujan’s death, Hardy had written Dewsbury: “Is it possible that Madras would consider the question of publishing the papers in a collected form? There should be some permanent memorial of so remarkable a genius; and this memorial would certainly be the most appropriate form.” Finally, in 1927, after protracted correspondence, Cambridge University Press came out with Ramanujan’s Collected Papers, 355 pages with almost everything he had ever published. The early Indian work was there. So was the partition function, and highly composite numbers, even questions he had posed readers of the Journal of the Indian Mathematical Society and the mathematical parts of his letters to Hardy.

  And with its publication, as the wider mathematical world took notice of Ramanujan’s work, the floodgates opened. Over the next few years, paper followed paper, dozens and dozens of them, with titles like “Two Assertions Made by Ramanujan,” and “Note on a Problem of Ramanujan,” and “Note on Ramanujan’s Arithmetical Function τ(n).” In 1928, Hardy handed over to G. N. Watson Ramanujan’s notebooks along with other manuscripts, letters, and papers, and he and B. M. Wilson set out on the mathematical adventure—editing the notebooks—that was to consume Watson until the eve of World War II and would result in more than two dozen major papers.

  So far as the mathematical community was concerned, Ramanujan lived.

  “During our generation no more romantic personality than that of Srinivasa Ramanujan has moved across the field of mathematical interest,” wrote American number theorist Robert Carmichael in 1932. “Indeed it is true that there have been few individuals in human history and in all fields of intellectual endeavor who draw our interest more surely than Ramanujan or who have excited more fully a certain peculiar admiration for their genius and their achievements under adverse conditions.”

  Writing in a kindred vein a few years later, Mordell wrote that “few other mathematicians for some generations past have been so full of human interest. The story of his life is that of the rise of an obscure Indian in the face of the greatest difficulties to the position of the most famous mathematician that India has ever produced and of his early death just after he had won the most coveted distinctions.”

  But if Ramanujan’s life exerted a peculiar hold on mathematicians, much more so did his work. In Hungary, in 1931, an eighteen-year-old University of Budapest prodigy, Paul Erdos, had written a paper on prime numbers. His teacher suggested he read a similar proof in Ramanujan’s Collected Papers, “which I immediately read with great interest.” Then, the following year, he saw a Hardy-Ramanujan paper concerned with the number of prime factors in an integer.

  Some numbers, recall, are more composite than others, a subject Ramanujan had explored in his longer paper on highly composite numbers. A number like 12 = 2 × 2 × 3 has more prime factors, three, than a number like 14 = 7 × 2, which has only two. The number 15 has two, while 16 has four. As you test each integer, the number of its divisors varies considerably. Well, Hardy and Ramanujan had said, we will examine not how this number varies but seek its average value—in the same way that you cannot predict the next throw of the dice yet can predict, on average, how often particular dice combinations will appear. Their result, loosely speaking, was that most integers have about log log n prime factors: note n, take its logarithm, then take its logarithm, and you wind up with a crude estimate that improves as n increases. But as rough as the result was, it beat anything anyone had before, and took twenty-three pages of close-grained mathematical reasoning to prove it.

  For almost twenty years, Hardy later told Erdos, their theorem seemed dead in the water, no progress being made in improving it. Then, in 1934, the problem was resurrected, and in 1939 Erdos and Mark Kac were led to a theorem that took it much further. It was only then that mathematicians could look back and pronounce the Hardy-Ramanujan paper of 1917 the founding document of the field that became known as probabilistic number theory.

  • • •

 
; In Norway, in 1934, a schoolboy named Atle Selberg, who was to become one of the world’s most famous number theorists, came upon an article about Ramanujan in a Norwegian mathematical journal. The article termed Ramanujan “a remarkable mathematical genius,” but as Selberg told an audience in Madras years later, the Norwegian word most aptly translated as “remarkable” also had connotations of “unusual and somewhat strange.”

  The article, with some of the results it included, “made a very deep and lasting impression on me and … fascinated me very much.” Then Selberg’s brother, also flirting with mathematics, brought home a copy of Ramanujan’s Collected Papers. To Selberg, this was “a revelation—a completely new world to me, quite different from any mathematics book I had ever seen—with much more appeal to the imagination.” Over the years, it retained its excitement and air of mystery. “It was really what gave the impetus which started my own mathematical work.”

  Later, his father gave him as a present his own copy of the Collected Papers which, as he told the Bombay audience, he carried with him still.

  • • •

  In England, in 1942, Freeman Dyson, a second-year student at Hardy’s alma mater, Winchester, won a school mathematics prize and selected a book on the theory of numbers by Hardy and E. M.Wright. “The chapter in Hardy and Wright which I loved the most,” he would recall many years later, “was chapter 19 with the title ‘Partitions,’ ” and featuring the congruence properties of the partition function discovered by Ramanujan. Dyson was intrigued, speculated on what the congruence properties might imply, and conceived the notion of “rank.” The rank of a partition, as he defined it, was its greatest part minus the number of its parts. Thus, one partition of 9 is

  6 + 2 + 1 = 9.

  Its “rank” is 3—the largest part, 6, less the total number of parts, 3. What Ramanujan’s congruence properties implied, speculated Dyson, was that certain partitions broke down neatly into equal-sized categories based on their rank.

  “That was the wonderful thing about Ramanujan,” he would say later. “He discovered so much, and yet he left so much more in his garden for other people to discover. In the forty-four years since that happy day, I have intermittently been coming back to Ramanujan’s garden. Every time when I come back, I find fresh flowers blooming.”

  Two years later, Dyson was working for the Royal Air Force Bomber Command as a statistician, where he saw up close, as all the Allied propaganda could not deny, the staggering losses over Germany. “It was a long, hard, grim winter,” he wrote later. In the evenings, he corresponded with another mathematician on ideas first advanced by Ramanujan. “In the cold dark evenings, while I was scribbling these beautiful identities amid the death and destruction of 1944, I felt close to Ramanujan. He had been scribbling even more beautiful identities amid the death and destruction of 1917.”

  • • •

  Over the years, then, Ramanujan was never forgotten. A 1940 listing noted 105 papers devoted to his work since his death. In the late 1950s, when Morris Newman delivered a paper at an Institute in the Theory of Numbers conference at the University of Colorado at Boulder, he began: “As with so much in analytic number theory, the study of congruence properties of the partition function originated with Ramanujan,” and used Ramanujan’s early papers as a jumping-off point. And certainly by the time of the centennial of his birth in 1987, Ramanujan’s reputation was secure.

  In his Introduction to the History of Mathematics, Howard Eves fashioned an outline chronicling the seminal moments in mathematics through the ages. The year 1906, for example, marked the work of Fréchet, the year 1907 that of Brouwer. Then, by Eves’s reckoning, came a long dry spell, with nothing, for nine years, that in the sweep of mathematical history ranked sufficiently high to include. In 1916, the dry spell was over with Einstein’s general theory of relativity. Then, for 1917, came Eves’s next entry:

  Hardy and Ramanujan (analytical number theory)

  Still, through most of the middle years of the century, a sense of tragedy and unfulfilled promise clung to Ramanujan’s name, of regret that he had not been greater still. Hardy himself set the tone, pointing to how Ramanujan had inhabited a mathematical desert for so many years. “He would probably have been a greater mathematician if he had been caught and tamed a little in his youth,” he wrote; “he would have discovered more that was new, and that, no doubt, of greater importance.” Littlewood echoed that sentiment in reviewing the Collected Papers in 1927: “How great a mathematician might Ramanujan have been 100 or 150 years ago? What would have happened if he had come into touch with Euler at the right moment?”

  But he hadn’t, of course; it was a shame, and that was that.

  For some years, his work went into eclipse, as new areas of mathematics, wholly distant from those Ramanujan had pursued, became fashionable. The Collected Papers, while it had its disciples, was no best-seller, even by scholarly standards. Just 42 books were sold the first year, 209 the second, and an officer of Cambridge University Press predicted in a letter to Hardy late in 1929 that it would be another ten years before the first printing, of 750, was sold out. “When I came to the United States [after World War II],” Freeman Dyson would recall, “I was all by myself as a devotee.” To the mathematical avant-garde of the day, Ramanujan’s work was no more than a “backwater,” a vestige of the nineteenth century.

  But that, in time, would change.

  5. RAMANUJAN REBORN

  One milestone came in the late 1950s. S. Ramaseshan, whose father had known Ramanujan, was visiting friends in Bombay, when he was taken to an inner room of their printing plant. There, he was

  shown a stack of browned old paper with magic squares and beautiful mathematical formulae systematically written in an elegant hand. I could not believe that I was face to face with Ramanujan’s notebook, the one I had heard about first from my father in 1937, the famous “frayed notebook” of Ramachandra Rao. I ran the tips of my fingers gently over the old paper—to feel the sheets which Ramanujan himself had filled with a smile on his face when he was without a job and everything else in his life seemed so bleak.

  His friends at the Commercial Printing Press had been given “one of the most exciting jobs they had ever undertaken.” It was 1957 and the Tata Institute of Fundamental Research, in a publishing venture financed by the Sir Dorabji Tata Trust, was bringing out Ramanujan’s notebooks in a facsimile edition of two physically daunting volumes that together weighed in at more than ten pounds.

  Until then, only Ramanujan’s published papers, along with his letters to Hardy, had been published. But those led Littlewood to think, when he reviewed them in 1929, that “the notebooks would give an even more definite picture of the essential Ramanujan.” The published papers mostly showed Ramanujan burnished by Hardy’s editing, his work all gussied up and tied in a ribbon. The notebooks, on the other hand, were Ramanujan in the raw.

  • • •

  On October 8, 1962, a group of men met at the three-hundred-year-old Mallikeswarar Temple, at the northern end of Linghi Chetti Street, in Madras’s Georgetown district, whose streets Ramanujan had walked half a century before. Here, in the shadow of the temple’s ornate gopuram, P. K. Srinivasan, a mathematics teacher at Muthalpiet High School, brought his friends together to launch a project. He had first read about Ramanujan twenty years before. Ever since, he had tried to inspire students with his example. Then, eight years before, a friend had taken him to meet Janaki and Tirunarayanan, Ramanujan’s surviving brother. Now, as the seventy-fifth anniversary of Ramanujan’s birth approached, he was determined to bring out a memorial book, filled with letters and reminiscences, to honor him.

  Recruiting the high school’s alumni, or “old boys,” to help him, he placed ads in local papers, interviewed people who had known Ramanujan, gathered letters. When he’d get some flicker of interest from an ad or contact, he’d immediately follow up. Often, he found himself just patiently sitting there, while someone rummaged around in an old trunk for some half
-remembered letter. Sometimes he’d bring in a stenographer, skilled in both English and Tamil, to record the conversation.

  Ramanujan’s seventy-fifth birthday was observed across South India. Town High School, in Kumbakonam, named one of its buildings after him. A stamp was issued in his honor; two and a half million copies of his passport photo, reduced to inch-high form, colored sienna, and valued at fifteen new paise, sold out the day they were issued. In Madras, around the time of the anniversary, a birthday celebration was held in his honor, and many of those who had been close to him or his family were in town. Srinivasan exploited the opportunity, stationing old boys at the entrance of the hall to solicit comments, correspondence, and reminiscences.

  At another point, he visited Ramanujan’s old house in Kumbakonam; Tirunarayanan had given him permission to look through the almirah, a sort of wardrobe, kept in a separate locked room of the house. In the presence of the tenant, Srinivasan unlocked the room. When he opened the almirah, which was covered with dust and cobwebs, cockroaches swarmed out. But in it, despite Tirunarayanan’s assurance that any such find was unlikely, he found a letter Ramanujan had written his father from England.

  P. K. Srinivasan’s compilation of letters and reminiscences came out in 1968. Brief biographies of Ramanujan appeared, in English, in 1967, 1972, and 1988; in Tamil in 1980 and 1986; and in Hindi, Kannada, Malayalam, among other Indian languages.

  • • •

  Then, in 1974, Deligne proved the tau conjecture.

  Almost sixty years before, in 1916, Ramanujan had published a paper with the unprepossessing title, “On Certain Arithmetical Functions.” An arithmetical function is one that originates in trying to learn certain properties of numbers; pi (n), the number of prime numbers, and p(n), the number of partitions, both problems on which Ramanujan worked, were two of them. The tau function, τ (n), was another.