My Search for Ramanujan

by Ken Ono

  It is true that there were mathematicians who might have recognized Ramanujan’s talent and creativity in the theory of modular forms and theta functions. However, I doubt that any of them would have been inspired to calculate partition numbers, a problem that deeply interested Hardy. Had Ramanujan had such a mentor, then analytic number theory would have missed out on one of its most important advances, the development of the “circle method.” Moreover, had Hardy and Ramanujan never done their work on partitions, then Rademacher would have had nothing to perfect, and the theory of Rademacher sums wouldn’t exist. And without the theory of Rademacher sums, much of the current work in mathematical physics wouldn’t exist.

  The questions that I really wonder about are these: What would Ramanujan have discovered with the help of a computer? What would he have discovered had he not died tragically at the age of thirty-two?

  Alas, the world will never know.

  Was Ramanujan the greatest mathematician of his time—or perhaps all time? My answer is best framed in the context of the lovely article “The two cultures of mathematics,” by Timothy Gowers, a recent British Fields medalist and current professor and Fellow of Trinity College. In that article, Gowers argues that there are really two kinds of mathematicians, generally speaking: those that are “problem solvers” and those that are “theory builders.” Gowers gives two examples to which he thought most mathematicians could relate. The first is Paul Erdős, who with no real home, famously traveled the world, living from his suitcase, working with other mathematicians. Erdős was a problem solver, and his many papers, written with over five hundred collaborators, are solutions to problems, stated, of course, as theorems or other mathematical results. Erdős was so prolific, and published so many papers with so many collaborators around the world, that his colleagues honored him by assigning an “Erdős number” to mathematicians based on the length of the chain of collaboration. Those who collaborated directly with Erdős have Erdős number 1; those who did not collaborate directly but collaborated with those collaborators have Erdős number 2, and so on, in a sort of “six degrees of separation” math game.

  As for the other type of mathematician, Gowers gives the example of Sir Michael Atiyah, of Oxford. It is not that Atiyah didn’t solve any problems—of course he did, and the famous Atiyah–Singer theorem named for him could be viewed as a problem that the two mathematicians solved. But Atiyah’s purpose as a mathematician, as Gowers shows from articles and interviews, was to pursue a theory—to stretch and test mathematics to its limits and then to extend those limits to build more mathematical structures. To be sure, there is an overlap, and theory builders also solve problems, and problem solvers, through their solutions, also extend our knowledge as a whole, and hence build theory. But based on their entire output, Atiyah and Erdős can serve as representatives of the two groups.

  Alexander Grothendieck is probably the best recent example of a theory builder. With his students, he rebuilt the field of algebraic geometry. He cared so little for details that he was famously quoted as once saying, when someone interrupted a talk he was giving asking for an example of a prime number, “Well, take 57.” Of course, 57 is not prime, since it equals 3 × 19. Andrew Wiles, of course, could be called a master problem solver for solving the greatest unsolved problem of all time: Fermat’s last theorem. But in contrast to the problems solved by Erdős, a behemoth of a theory underlies Wiles’s solution, namely the Langlands program, which in a sense is unifying major branches of mathematics.

  Ramanujan could well be viewed as a problem solver, because much of his work consisted in actually solving a huge number of extremely complicated problems in mathematics. But there are theories lurking behind those problems, and the problems are part of the theory. Therefore, in Ramanujan’s case we should resist classifying him as either a problem solver or a theory builder.

  In his biography of Ramanujan, Kanigel uses the following analogy to describe Ramanujan’s prowess as a mathematician:A car mechanic reliant on mechanical instinct may “know” how an engine works yet be unable to set down the physical and chemical principles governing it.

  Because of his lack of formal training and the fact that he was set in his mathematical ways by the time he reached Cambridge, Ramanujan could not be a theory builder, simply because he didn’t know enough about how mathematics worked as an edifice. But his intuition was supreme—100 on a scale of 0 to 100 that Hardy had suggested, in which he gave himself a mere 25, and the great David Hilbert an 80. His immensely powerful intuition, then, allowed Ramanujan to propose and sometimes prove very difficult and unexpected problems in mathematics, but he was not a Grothendieck—someone who built theories with little concern for individual problems.

  So here is my answer to the question whether Ramanujan was the greatest mathematician of his time—perhaps of all time. Ramanujan was a gift; he should be remembered as the greatest anticipator of mathematics. Although he was recognized during his lifetime, his most important ideas, those that have powered mathematicians after his death, were largely viewed as insignificant while he was alive. Those ideas are typically found in his notebooks, letters, and papers as innocent-looking formulas and expressions. Those gems have offered visions of the future, hints of subjects conjured long after his death. For mathematicians, theorems and proofs are works of art, and these formulas are reminiscent of the masterpieces by the Dutch artist Vincent van Gogh, who died at the age of thirty-seven, before his work was fully appreciated.

  Ramanujan’s formulas have inspired many influential mathematicians, such as Andrews, Deligne, Dyson, Selberg, Serre, Weil, to name just a few, and they have supplied prototypes of deeper objects in algebraic number theory, combinatorics, and physics. Ramanujan was an incredibly great mathematician, certainly up there with the greatest in history. He had a gift of imagination the like of which the world of mathematics had never seen before.

  Fermat’s Last Theorem and the Tokyo–Nikko Conference

  The 1955 Tokyo–Nikko meeting, which was a pivotal event in my father’s life, turned out to be a pivotal event in the history of mathematics. At that conference, Yutaka Taniyama, one of my father’s close friends, posed a problem that suggested a deep connection between seemingly unrelated objects: “modular forms” and “elliptic curves.” This problem would evolve over time, and its final form became known as the Taniyama–Weil conjecture, the Shimura–Taniyama–Weil conjecture, and the modularity conjecture. Goro Shimura, who also attended the symposium, was a star, a leader among the young ambitious Japanese mathematicians.

  Although number theorists understood the importance of the modularity conjecture early on, it was the work of Berkeley mathematician Ken Ribet in the late 1980s that catapulted the conjecture to prominence. Ribet proved that the conjecture implies Fermat’s last theorem.

  Ribet’s work provided a new approach to Fermat: prove the modularity conjecture. Although mathematicians were excited to learn of the deep connection between the modularity conjecture and Fermat’s last theorem, few believed that it would lead anywhere. The result was viewed as further evidence that both problems would be difficult to solve, that they might remain unresolved for generations. But then in 1993, Andrew Wiles made world news when he announced that he had proved Fermat’s last theorem. That proof made use of Ribet’s theorem.

  Ken Ribet, Peter Sarnak, Ken Ono in 2014 (photo by Ling Long)

  In this way, the proof of Fermat’s last theorem, one of the most important events in the history of mathematics, was born from humble beginnings: an innocent question raised by an unproven young Japanese mathematician at a conference intended to promote reconciliation and world peace in the name of science.

  Tragically, Taniyama did not live to enjoy the glorious work that sprouted from his conjecture. Suffering from depression and a loss of confidence, he committed suicide on November 17, 1958. One month later, his fiancée, Misako Suzuki, also committed suicide, and she left behind a note in which she wrote, “We promised each other
that no matter where we went, we would never be separated. Now that he is gone, I must go too in order to join him.”

  Their deaths are tragic, but in a way, their suicides may have made sense to their families and friends. In Japanese culture, there is a long history and tradition of suicide. There is the concept of an honorable suicide, such as seppuku practiced by the samurai; the crashing of one’s plane into enemy ships as practiced by World War II kamikaze pilots; and suicide intended to prevent shame from befalling one’s family. And Misako Suzuki’s suicide is part of a long tradition in Japan of shinjū, or love-suicide, the unhappy lovers believing that they will be reunited in heaven.

  Mathematical Gems

  This book would not be complete without a discussion of some of Ramanujan’s works. Here we include a small sample of Ramanujan’s mathematical ideas and conjectures.

  The Hardy–Ramanujan Taxicab Number

  While Ramanujan was lying ill, this time at a hospital in Putney, just outside London, Hardy would come to see him by taking a train from Cambridge to London, and then a taxi to Putney. And so one day, Hardy made the trip to see Ramanujan. In an attempt to lift the ailing mathematician’s spirits, Hardy led off with a casual comment about a number, numbers being Ramanujan’s favorite topic: “I came here in a taxi with a very dull number: 1729.” But to Ramanujan, who was said to be a friend of every integer, that number wasn’t dull at all. To Hardy’s surprise, Ramanujan gathered what strength he had, jumped up in bed, and cried, “No, Hardy—it is a very interesting number! It is the smallest number expressible as the sum of two cubes in two different ways!” (). From this event, the mathematical study of taxicab numbers—the smallest numbers that can be expressed as the sum of two (positive) cubes in n distinct ways—emerged. To date, only six taxicab numbers have been identified.

  Ramanujan was aware of this property of 1729 because of work he had done on a problem studied by Euler that can be found in his notebooks. The number 1729 appears in Ramanujan’s works in yet another context, this time related to Fermat’s last theorem. It appears that he was thinking about near misses to Fermat’s claim. That is, the Fermat conjecture would be false if there existed a positive integer n (greater than 2) and nonzero integers x, y, and z such that x n  + y n  = z n . It turns out that Fermat’s last theorem comes “close” to being false—one unit short—for n = 3. The number 1729 is the sum of two cubes (powers of n = 3): 93 + 103, in one of the two possibilities. But 1729 is not itself a cube. If it were, Fermat’s theorem would be false. But 1729 “misses” being a cube by only one unit, since 1728 is a cube: .

  Two years ago, I visited the Wren Library at Trinity College, where I studied the Ramanujan archives. To my surprise, I found a page in Ramanujan’s handwriting in which he lists families of near misses to Fermat’s last theorem, and one he offers is . It turns out that he had discovered identities that mathematicians would later find to be important in algebraic geometry and mathematical physics. Together with my PhD student Sarah Trebat-Leder, we have discovered that these identities can be reformulated as statements about K3 surfaces and ranks of elliptic curves, two important subjects that did not exist in Ramanujan’s day.

  Ramanujan’s page on sums of two cubes and near misses to the cubic Fermat equation (photo courtesy of Trinity College)

  Ramanujan derived so many formulas and theorems in Cambridge that it has taken mathematicians decades to resolve his findings. This book would not be complete without discussing some of his results. Here we offer a glimpse of some of his most well known and influential works.

  Approximations to π

  Early in their collaboration, Hardy and Ramanujan went over the claims that so densely filled his letters and the notebooks that Ramanujan had brought to England. In all, there were three or four thousand such results!—filling pages and pages.

  The year Ramanujan arrived in England, his first paper was published in an English journal, the Quarterly Journal of Mathematics. It was titled “Modular Equations and Approximations to Pi,” and it included amazing formulas for

  well known to every schoolchild as the ratio of the circumference of a circle to its diameter. The number π is an example of an irrational number, a number that cannot be described as a simple ratio of integers. One consequence of its irrationality is that its decimal expansion is infinite, running on forever without a discernible pattern, in contrast to the pattern that can be seen in the decimal expansion of the rational number

  Amateur and professional mathematicians alike have been captivated by the decimal expansion of π. Despite the fact that π is easily described as a simple ratio in a circle, it is not so simple to calculate its decimal places.

  I find it incredible that in 2013, ninety-nine years after Ramanujan wrote his first paper with Hardy, the Russian-American brothers David and Gregory Chudnovsky used a home-built supercomputer running a variant of Ramanujan’s formula to determine the first 12.1 trillion digits of this number (with earlier estimates of ten trillion digits of π in 2011 and five million digits in 2010). Their algorithm uses a very rapidly converging infinite series that is a relative of Ramanujan’s original formula:

  This work opened the way to very similar, yet more and more accurate, ways of obtaining more and more digits of π.

  Highly Composite Numbers

  During his second year in England, Ramanujan produced many more new and original papers—some with Hardy, some that were all his own. As Hardy described it, Ramanujan’s “flow of original ideas showed no symptom of abatement.” Ramanujan worked on the distribution of prime numbers, the Riemann hypothesis, prime factorization of integers, and more. In 1915, Ramanujan defined a new concept: a highly composite number.

  Integers are the basic building blocks of mathematics, and prime numbers are the basic building blocks of the integers, for every integer can be broken down, multiplicatively, uniquely as a product of primes. For example, 6 = 2 × 3 and 15 = 3 × 5 and . Using this multiplicativity, one can list all the divisors of a given number. The only divisor of 1 is 1. The number 2 has two divisors, namely 1 and 2. The number 12 has six divisors, namely 1, 2, 3, 4, 6, 12.

  A highly composite number is one that has more divisors than any smaller positive integer. The first few highly composite numbers are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120. Incidentally, one theory in the history of mathematics, proposed by the Austrian-American scholar Otto Neugebauer, is that the Babylonians chose base 60 for their number system because a base with a large number of divisors makes it easier to do arithmetic. Ramanujan defined and studied these highly composite numbers, and he studied their frequency.

  Euler’s Partition Numbers

  Ramanujan was a pioneer in a subject that is very dear to me, that of looking at the integers as sums rather than products. The partition numbers are the numbers we discussed earlier that show the number of ways in which an integer can be broken down as a sum. To recapitulate from an earlier chapter: The equalities 3 = 2 + 1 = 1 + 1 + 1 illustrate that there are three ways of “partitioning” the number 3. Next we observe that 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1, which shows that there are five ways of partitioning the number 4. Repeating this process of adding and counting for every number n defines the partition function p(n). Thus our examples can be denoted by p(3) = 3 and p(4) = 5.

  As we mentioned before, the partition numbers grow at an astonishing rate. You might not think so from calculating p(10) = 42, p(20) = 627, and p(30) = 5604. But there are nearly four trillion ways of partitioning 200. Obviously, it would be crazy to attempt to list all the partitions of 200 and then count them one by one. And even if you could, by the time you get to 1000, you would have to deal with p(1000) ≈ 3.61673 × 10106. Compare this with the total number of atoms in the known universe, which is “only”—at the highest estimate—4 × 1081.

  There must be a better way to determine such numbers.
>
  The great Leonhard Euler studied this problem in the eighteenth century, and he found a clever way of computing partition numbers that avoids the impossible task of counting the partitions one by one. He found a “recurrence relation,” a procedure that computes these numbers in order. For example, his method makes it possible to compute p(200) if one has prior knowledge of the numbers p(0) = 1, p(1) = 1, p(2) = 2, … , p(199). Euler’s procedure was a major improvement, but it is quite cumbersome.

  Ramanujan wanted a better result: he wanted a formula for the partition numbers that would merely require plugging in for n to get p(n). Working with Hardy, Ramanujan came close to finding such a formula. Hardy and Ramanujan invented the “circle method,” a device in analytic number theory (a branch of number theory that uses methods from mathematical analysis) that has become one of the most important tools in mathematics, to obtain an amazing approximation for the partition numbers. They proved the “asymptotic formula”

  into which one can plug in for n and get back a number that is reasonably close to the partition number p(n). Here e is Euler’s number e = 2.718… . This formula predicts that there should be 199,280,895 ways of partitioning 100. It turns out that the partition number for 100 is actually 190,569,292, which is less than five percent below the approximate value. The Hardy–Ramanujan formula for 500 gives a prediction that is off by less than two percent, and by 5000, their formula is off by less than one percent. Although the asymptotic formula never gives the exact answer, the percentage error shrinks quickly for larger and larger numbers. After Hardy and Ramanujan produced this astonishing approximate formula, it would be many years before significant progress would be made on the problem of computing the partition numbers exactly.