The Man Who Knew Infinity

by Robert Kanigel

  “Let σs(n) denote the sum of the sth powers of the divisors of n,” Ramanujan had begun. If n = 6, for example, its divisors are 6, 3, 2, and 1. So that if, say, s = 3, “the sum of the sth powers of the divisors,” σ3 (6), is just 63 + 33 + 23 + l3 = 252. But how calculate σs(n) generally? That question led Ramanujan, after fifteen pages, to the tau function, whose properties, Hardy would write twenty years later, “are very remarkable and still very imperfectly understood.” As in so much of analytical number theory, stepping through the open door of a simple-seeming problem had led into a mathematical labyrinth of formidable complexity.

  Ramanujan’s hypothesis, as Hardy called it, or the tau conjecture as it became more generally known, did not offer an explicit formula for τ (n). Rather, it merely stated that τ (n) was “on the order of …” something. In other words, like the prime number theorem, it was a kind of approximation. “There is reason for supposing,” Ramanujan wrote, “that τ (n) is of the form 0 (n11/2+∈).” The notation was a way of saying that its value, whatever it actually was, was always less than something. What was “something”? Leaving out constants, it was, at most, n to some power only slightly greater than 11/2 (which means the square root of n to the eleventh power).

  Ramanujan’s conjecture, in the words of S. Raghavan of the Tata Institute of Fundamental Research in Bombay, “kept at bay a whole galaxy of distinguished mathematicians for nearly six decades,” remaining one of the major open problems in number theory. An account by Hardy in 1940 hinted at both the interest it generated and its resistance to solution. Ramanujan himself, he reported, had proved that τ (n) = 0(n7); but 7, the power of n here, was much more than 11/2, and so still far short of proving the tau conjecture. Two years later, Hardy himself cut a little closer, to 0(n6). Kloosterman got closer in 1927, Davenport and Salie closer still in 1933. The Scottish mathematician Robert Rankin, a student of Hardy’s, proved in 1939 that τ (n) = 0(n29/5).

  All this was a little like showing, first, that the murderer lived at a house on Union Street numbered less than 2170; then, that the house number was less than 2160; then, less than 2158 … And yet still you couldn’t prove he lived at 2155, which is where your prime suspect lived.

  It was only in 1974 that the Belgian mathematician Pierre Deligne, in what would be described as “one of the celebrated events of 20th century mathematics,” proved the conjecture using powerful new tools supplied by the field known as algebraic geometry. Deligne was subsequently awarded the Fields Medal, the mathematical community’s counterpart to the Nobel Prize, for his triumph.

  And when he did it, it helped solidify Ramanujan’s reputation all the more.

  • • •

  In 1988, looking back upon the fall and rise of Ramanujan’s reputation over the years, Bruce Berndt compared him to Johann Sebastian Bach, who remained largely unknown for years after his death in 1750. For Bach, the big turnaround came on March 11, 1829 with Felix Mendelssohn’s performance of the St. Matthew Passion. For Ramanujan, suggested Berndt, the roughly analogous event was George Andrews’s discovery of the Lost Notebook in 1976.

  In late April of that year, Andrews was a young University of Wisconsin visiting professor bound for a one-week conference in France. The airline fare structure made it cheaper to go for three weeks than one, so he looked for something to keep him, his wife, and his two daughters occupied during the extra time. A side trip to Cambridge? A colleague there had suggested he might be interested in rummaging through some papers left behind by G. N. Watson at his death in 1965. So Cambridge it was.

  Watson, who had worked on Ramanujan’s papers for many years before World War II, was a major classical analyst from the prewar period and a Fellow of the Royal Society. When he died, the society asked J. M. Whittaker, son of one of Watson’s collaborators, to write his obituary. Whittaker wrote Watson’s widow: could he come see his papers? She invited him to lunch, then took him upstairs to the study. There, Whittaker recalled, papers

  covered the floor of a fair sized room to a depth of about a foot, all jumbled together, and were to be incinerated in a few days. One could only make lucky dips [into the rubble] and, as Watson never threw away anything, the result might be a sheet of mathematics but more probably a receipted bill or a draft of his income tax return for 1923. By an extraordinary stroke of luck one of my dips brought up the Ramanujan material.

  This “material,” all 140 pages of it, was part of a batch of papers Dewsbury had sent Hardy in 1923 that had later gone to Watson. After his “lucky dip,” Whittaker passed them on to Robert Rankin, who in 1968 handed them to Trinity College.

  Whittaker and Rankin, both professional mathematicians of high standing, but whose backgrounds ill-suited them to distinguish this material from what had already appeared in the published notebooks, had failed to see in it what Andrews saw now. Within a few minutes, he realized that some of it bore on mock theta functions, the subject of his own Ph.D. thesis, and related subjects. Which meant these were papers Ramanujan could have generated only during the last year of his life in India.

  He was thrilled, “extremely excited that I had my hands on something spectacular.” But what now? Get the papers photocopied, of course.

  The interior of the Trinity College library is a visually arresting place that owes its beauty to the great English architect Christopher Wren. After its completion in 1695, someone said of the Wren Library that it “touches the very soul of any one who first sees it.” Everywhere are delicate woodcarving, stained-glass windows, and statues filling apses on the wall. Busts of Cambridge greats are arrayed along it length, and Thorwaldsen’s statue of Lord Byron stands at the far end. It is at a long table behind the Byron statue that library staff seat scholars studying medieval manuscripts, or the papers of Newton, or the poems of Milton. And in 1976, George Andrews was one of them. To him, the Wren was “a shrine, with busts of Newton glowering down on you. The idea of going up to the desk and having something xeroxed terrified me.”

  He mustered up his courage and told them what he wanted. “That will be about seven pounds” for airmail postage to the United States, they warned him. “Will that be all right?”

  It would be fine, he assured them. “I was ready to take a second mortgage on my house to get it.”

  In a paper appearing soon after he had unearthed it, Andrews styled his find “the Lost Notebook.” Its discovery, mathematician Emma Lehmer was moved to say, was “comparable to the discovery of a complete sketch of the tenth symphony of Beethoven.”

  But the appellation ruffled feathers in Britain. Robert Rankin pointed out, for example, that it was not a “notebook” but loose papers; and that, pristinely secure within the walls of the Wren Library, it had never been lost. Still, as Andrews observed later, “the manuscript and its marvelous results disappeared from any mention or account by the mathematical community for more than 55 years.” In that sense, they had indeed been “lost.” His contribution lay “not in saving them from oblivion as [Whittaker and Rankin] did,” but in recognizing them for what they were.

  Sometime before, in January 1974, Bruce Berndt had been on sabbatical at the Institute for Advanced Study and came upon papers bearing upon his own work of two years before that proved some formulas in Ramanujan’s notebooks. The institute library had no copy of the facsimile edition, but neighboring Princeton’s did. In it, he found a slew of related formulas that seemed similar to the others but that, try as he might, he couldn’t prove. “And that,” he recalled, “bothered me.” He set an informal task for himself—to prove all the results in chapter 14. That took him a year. Later, Andrews visited Illinois and mentioned that Watson and Wilson had spent years trying to prove the theorems in the notebooks, and that their own notes were still around. Ever since, beginning in May 1977, Berndt has worked on editing Ramanujan’s notebooks, “and I haven’t done anything else since.”

  For the decade ending in 1988, a computer search of the literature revealed, some three hundred papers referred to Ramanujan
in their titles or their abstracts. Ties and cross-links to other areas of mathematics, what Freeman Dyson calls “connections to deep structure and more general abstract notions,” were showing up everywhere. Over the past twenty years or so, Dyson says, “It has become respectable again to take Ramanujan seriously. So much that he conjectured was not just pretty formulas but had substance and depth.” The great tree that was Ramanujan’s work sent its roots down deep and far.

  He had planted it for himself, not to better the material condition of India or of the world. And yet, its underground tendrils ranged into fields far distant from pure mathematics—and into applications which, Hardy might have cringed to learn, were by no means “useless.”

  6. BETTER BLAST FURNACES?

  “If I am asked to explain how, and why, the solution of the problems which occupy the best energies of my life is of importance to the general life of the community, I must decline the unequal contest,” wrote Hardy, just after the guns of World War I had stilled.

  I have not the effrontery to develop a thesis so palpably untrue. I must leave it to the engineers and the chemists to expound, with justly prophetic fervour, the benefits conferred on civilization by gas-engines, oil, and explosives. If I could attain every scientific ambition of my life, the frontiers of the Empire would not be advanced, not even a black man would be blown to pieces, no one’s fortunes would be made, and least of all my own. A pure mathematician must leave to happier colleagues the great task of alleviating the sufferings of humanity.

  Whether and how the engineers and chemists might indeed apply Ramanujan’s work to the common purposes of life strikes a sensitive chord in India, beset as it is by practical problems of great urgency and less naturally inclined to trust in research whose rewards may accrue only decades or centuries later. “Several theorems of Ramanujan are now being widely used in subjects like particle physics, statistical mechanics, computer science, cryptology and space travel in the United States—subjects unheard of during Ramanujan’s time,” The Hindu assured its readers in its December 19, 1987 issue. But efforts to justify Ramanujan’s work on utilitarian grounds go back almost to his lifetime.

  At the Third Conference of the Indian Mathematical Society, held in Lahore soon after his death, Ramanujan’s life and work were on every speaker’s lips. “Mention of Ramanujan’s name,” declared the society’s president, Balak Ram, “suggests the question of the organization and endowment of scientific research in India,” in particular the balance between applied and pure. The engineer might be rewarded for his inventiveness, become rich the way Edison did. But what of learning for learning’s sake? Ought society reserve a place for it? Yes, he insisted.

  Every settled community is subconsciously or consciously convinced that attempts at progress are apt to be wasteful unless guided by thinkers and teachers who have leisure to think and teach clearly; therefore, we find these communities giving special protection and advice to learned men and thus providing them with leisure and the psychological incentives which replace the missing pecuniary reward.

  If India was to be assured of “progress,” he was saying, it had to allow for its Ramanujans. In the long run, it would pay off.

  When the Ramanujan stamp came out in 1962, the Indian postal service took pains to point out the potential applications of his work: “His work and the work of other mathematicians on Riemann’s zeta function, done in another context, has now been geared to the technological mill. It has been applied to the theory of pyrometry, the investigation of furnaces aimed at building better blast furnaces.” And his work on mock theta functions, modular equations, and in other realms was being studied for its possible application to atomic research.

  Down through the years, Ramanujan’s mathematics has indeed been brought to bear on practical problems, if sometimes tangentially. For example, crystallographer S. Ramaseshan has shown how Ramanujan’s work on partitions sheds light on plastics. Plastics, of course, are polymers, repeating molecular units that combine in various ways; conceivably, you might have one that’s a million units long, another of 8251, another of 201,090, and so on. Ramanujan’s work in partitions—on how smaller numbers combine to form larger ones—plainly bears on the process. As it does, for example, in splicing telephone cables, where shorter subunits, of varying lengths, again add up to make a whole.

  Blast furnaces? Plastics? Telephone cable?

  Cancer?

  At a meeting of the southeastern section of the American Physical Society in Raleigh, North Carolina, in November 1988, three University of Delhi researchers presented a paper entitled “A Study of Soliton Switching in Malignancy and Proliferation of Oncogenes Using Ramanujan’s Mock-Theta Functions.” They were using Ramanujan’s mathematics to help understand cancer, if only as a small, tangential contribution to a vast and complex subject. When the Hindu noted the paper, however, it assigned the headline “Ramanujan’s Maths Help Fight Cancer.”

  In other fields, Ramanujan’s mathematics has played a more decisive role—as in, for example, string theory, which imagines the universe as populated by infinitesimally short stringlike packets whose movement produces particles. Grounded in the real world or not—the jury is still out—the mathematics required to describe these strings demands twenty-six dimensions, twenty-three more than the three on which we manage in everyday life. Partition theory and Ramanujan’s work in the area known as modular forms have proved essential in the analysis.

  An important problem in statistical mechanics has also proved vulnerable to Ramanujan’s mathematics—a theoretical model that explains, for example, how liquid helium disperses through a crystal lattice of carbon. As it happens, the sites helium molecules may occupy in a sheet of graphite, say, can never lie adjacent to one another. Since each potential site is surrounded by six neighbors in a hexagonal array, once it is filled, the six around it define an unbreachable hexagonal wall. Reviewing in 1987 the work for which he received the prestigious Boltzmann Medal seven years earlier, R. J. Baxter of the Australian National University in Canberra set out the thinking behind his “hard hexagon model.” Mathematically, he showed, it was built on a particular set of infinite series. And “these series,” he observed, “are precisely those that occur in the famous Rogers-Ramanujan identities” (though he hadn’t realized it back in 1979 when he’d found them on his own). Based on them, he found a way to determine the probability that any particular site harbored a helium molecule; predictions borne of the model agreed closely with experiment. “The Rogers-Ramanujan identities,” Baxter concluded, “are perhaps not so remote from ‘ordinary human activity’ as Hardy would have liked!”

  Computers, scarcely the dream of which existed in 1920, have also drawn from Ramanujan’s work. “The rise of computer algebra makes it interesting to study somebody who seems like he had a computer algebra package in his head,” George Andrews once told an interviewer, referring to software that permits ready algebraic manipulation. Sometimes in studying Ramanujan’s work, he said at another time, “I have wondered how much Ramanujan could have done if he had had MACSYMA or SCRATCHPAD or some other symbolic algebra package. More often I get the feeling that he was such a brilliant, clever and intuitive computer himself that he really didn’t need them.” Then, too, a modular equation in Ramanujan’s notebooks led to computer algorithms for evaluating pi that are the fastest in use today.

  • • •

  “Hype,” Freeman Dyson calls some of what he deems undue fanfare for practical applications of Ramanujan’s work. To apply his mathematics to string theory is “stretching the point,” he says. “You don’t have to read Ramanujan to do string theory.” It was all true enough, and completely valid as far as it went—but peripheral, if not irrelevant, to most mathematicians.

  What makes Ramanujan’s work so seductive is not the prospect of its use in the solution of real-world problems, but its richness, beauty, and mystery—its sheer mathematical loveliness. Hardy was enraptured by it. And Dyson was, and Selberg, and Erdos, and
many, many others. “The best seem to appreciate Ramanujan early,” says Richard Askey, a University of Wisconsin mathematician deeply involved in Ramanujan’s work. “The rest of us have to need some of his work before really appreciating it.” George Andrews once told an interviewer how, as a young man, he was beguiled by the Ramanujan-Hardy partition formula. “I was stunned the first time I saw this formula. I could not believe it. And the experience of seeing it explained, and understanding how it took shape … convinced me that this was the area of mathematics I wanted to pursue.”

  Because it lies on a cool, ethereal plane beyond the everyday passions of human life, and because it can be fully grasped only through a language in which most people are unschooled, Ramanujan’s work grants direct pleasure to only a few—a few hundred mathematicians and physicists around the world, perhaps a few thousand. The rest of us must either sit on the sidelines and, on the authority of the cognoscenti, cheer, or else rely on vague, metaphoric, and necessarily imprecise glimpses of his work.

  Some Ramanujan experts have resorted to the language of other, more accessible realms of knowledge to suggest his hold on the mathematical imagination.

  Emma Lehmer, recall, likened The Lost Notebook to a Beethoven symphony.

  Watson concluded his presidential address to the London Mathematical Society in 1937 by saying that one Ramanujan formula gave him “a thrill which is indistinguishable from the thrill which I feel when I enter the Sagrestia Nuova of the Capelle Medicee and see before me the austere beauty of the four statues representing, ‘Day,’ ‘Night,’ ‘Evening,’ and ‘Dawn’ which Michelangelo has set over the tombs” of the Medicis.

  Berndt compared Ramanujan to Bach and, alluding to Ramanujan’s devotion to mathematics, quoted from Shelley’s “Hymn to Intellectual Beauty”: