My Search for Ramanujan

by Ken Ono

  The important innovation that Hardy and Ramanujan brought to this study was a classic example of the field the two of them were championing: analytic number theory. It used a very powerful “continuous” method, Cauchy’s theorem, to attack a discrete problem having to do with counting.

  This work drew the attention of a particularly intriguing mathematician at Trinity named Percy Alexander MacMahon (1854–1929). MacMahon was known to everyone as “Major MacMahon.” The reason for this was that in the decade before Ramanujan was born, the 1870s, MacMahon had a brilliant military career. After graduating from the Royal Military Academy at Woolwich, MacMahon was sent as a military officer to India, and in an interesting coincidence, served in Madras. By the time Ramanujan was born, MacMahon was back in England, first as a military officer, but later also as a mathematician, working in the area of combinatorics.

  He did so well in mathematics that in 1890, he was elected a Fellow of the Royal Society (FRS), and eight years later, he retired from the military altogether to devote himself fully to mathematics. He became very interested in partitions. And he had an amazing skill: like John von Neumann many years later, MacMahon was a human “calculator.” He could perform very complicated calculations rapidly in his head.

  When he met Ramanujan at Cambridge and became aware of the Hardy–Ramanujan work on partitions of numbers, Major MacMahon offered his considerable calculating abilities to help the pair. He would simply compute the number p(n) for successively larger values of n, compiling large tables of exact values against which Ramanujan and Hardy could compare their own numbers, which they obtained from successively improved estimates of the function that they believed would lead to the “true” p(n).

  Ramanujan also proved the very surprising divisibility properties discussed earlier. Recall that he discovered that there exists an arithmetic progression all of whose partition numbers are divisible by 5, that is, that is always a multiple of 5. The sequence of these partition numbers begins p(4) = 5, p(9) = 30, p(14) = 135, p(19) = 490, p(24) = 1575, all multiples of 5. He also proved, as discussed earlier, a similar striking theorem for 7 and 11, namely that p(7n + 5) is always a multiple of 7, and p(11n + 6) is always a multiple of 11. These statements are now known as “Ramanujan’s partition congruences.”

  Hardy and Wright state in their textbook on number theory how those arithmetic properties of the partition counter p(n) were derived by Ramanujan:Examining MacMahon’s table of p(n), [he] was led first to conjecture, and then to prove, these striking arithmetic properties associated with the moduli 5, 7, and 11.

  Hardy and Wright then present Ramanujan’s brilliant proofs of these theorems, all of which exploit infinite series.

  The Rogers–Ramanujan Identities

  In Cambridge, Ramanujan returned to some of his earlier work from his days in India. He revisited identities that he knew were related to the last formulas in his first letter to Hardy, the ones that Hardy said had defeated him completely. These are the formulas that Hardy believed had to be true because “if they weren’t, nobody would have had the imagination to invent them.” These formulas are close to my heart; their present-day formulation is

  While he was seeking to prove these identities, Ramanujan discovered that someone else had already discovered them. It was a mathematician by the name of Leonard James Rogers (1862–1933), at the University of Leeds, who also had the prestigious distinction of being a Fellow of the Royal Society, but had the unfortunate knack for having his work ignored by the rest of the world.

  It was in 1917 that Ramanujan discovered, in an 1894 issue of the Proceedings of the London Mathematical Society, where it had gone unnoticed for twenty-three years, Rogers’s paper on the exact same identities that he had found! Ramanujan contacted Rogers, presumably with Hardy’s help, and the two men then worked together and provided a new, joint proof of these relations, which became known as the Rogers–Ramanujan identities.

  Unbeknownst to them, the mathematician Issai Schur (1875–1941), an expert in the mathematical area called representation theory working in Berlin, proved these same identities in 1917. The Rogers–Ramanujan proof would appear only two years later, in 1919.

  What each of the two identities does is to give us a function, whose variable is q, represented (for each equation) in two different ways. On the left is an infinite sum of fractions involving the variable q, and on the right is the reciprocal of an infinite product of terms involving q. Even Hardy, in his book on number theory with Wright, notes that the proofs are not simple. But we can say something about what these identities mean. Each side of the equation can be expanded as a formal power series. To begin with, each expression in the identity is a slight variation of the function f(x) = 1/(1−x), which we can express as a formal power series as follows::

  This is the standard “geometric series” that one encounters in calculus. We can now, for example, express the left-hand side of the first identity above as follows:

  And expanding the right-hand side gives exactly the same terms! The fact that the terms match up is evidence of a theorem, but it is not a proof: how do we know that all the terms match, right out to infinity? The actual proof in this case requires deep ideas that involve some tricky manipulation of power series.

  These two identities were the secret behind Ramanujan’s stunning expressions, which we discussed earlier in connection with the golden ratio ϕ, which had so astonished Hardy.

  Ramanujan’s “imagination,” the like of which nobody else possessed, was that he figured out how to use these two identities together to magically produce examples of numbers like the golden ratio ϕ by choosing for the variable q the “crazy” numbers −e −π and e −2π in his continued fraction

  Hardy didn’t know, indeed nobody knew, how Ramanujan had done it.

  Not only did Ramanujan figure out these numbers, it turned out that his expressions involved only “algebraic numbers,” that is, numbers that are roots of polynomial equations with integer coefficients. This is an important class of numbers in mathematics. It includes the golden ratio ϕ, since it is a solution to . In some sense, “most” numbers are not algebraic. The numbers e and π, for example, are not. Since algebraic numbers are relatively rare, it is amazing that the two wild examples from Ramanujan’s first letter that defeated Hardy completely,

  correspond to solutions to the single polynomial equation .

  Ramanujan’s Tau Function

  Ramanujan’s innocently titled paper “On certain arithmetical functions” is arguably his most important work. The results in this paper include “congruences,” which would later play a role in developing ideas that were critical to the proof of Fermat’s last theorem, and it contained a conjecture that would later be proved by Deligne in his 1974 paper for which he was awarded the Fields Medal. Although it is too ambitious to try to explain these results in detail, it is worthwhile to see glimpses of what Ramanujan did.

  Ramanujan defined a function that he denoted by the Greek letter tau (τ). It is obtained by multiplying infinitely many polynomials together. Now, that is exactly what is done on the right-hand side of the Rogers–Ramanujan identities shown above, where you can see an infinite product of polynomials in the denominator. Now, the only way for such a product to make sense is for the terms representing successively higher powers of the variable to stabilize. And that will happen only if any given power of the variable appears in only finitely many of the polynomials. Those conditions are satisfied by Ramanujan’s function. That allows us to get more and more terms of the function by multiplying out more and more of the polynomials, as follows:

  As one multiplies more of these polynomials, the terms at the beginning stabilize. Note that the second and third polynomials share the first three terms x−24x 2 + 252x 3, and every new polynomial from then on will begin with these three terms. If we were to multiply the next polynomial, then all the polynomials from then on would share the first four terms, and so on. In this way,
Ramanujan was able to define a single object, a “power series” formed by multiplying infinitely many polynomials together. It is his “Delta” function

  The tau function is now defined on the positive integers by taking the coefficients of the Delta function in order: τ(1) = 1, τ(2) = −24, τ(3) = 252, τ(4) = −1472, and so on.

  Ramanujan proved that if p is prime, then −1−p 11 + τ(p) is without exception a multiple of 691. Here we show this for the first few small primes:

  As strange as this may seem, it turns out that this phenomenon is a prototype of one of the deepest theories to be developed in the second half of the twentieth century, the theory of Galois representations, a universe that was imagined by Évariste Galois. How this leads to the proof of Fermat’s last theorem is one of the longest and most beautiful adventures in the history of mathematics.

  Ramanujan conjectured that his tau function is multiplicative, meaning that if m and n share no common prime factors, then τ(mn) = τ(m)τ(n). For example, if m = 2 and n = 3, then using the numbers above, we find that τ(2)τ(3) = −24 × 252 = −6048 = τ(6). Ramanujan didn’t prove this conjecture; it would later be proved by Louis Mordell, and it would go on to serve as the prototype of a central feature of modular forms, among the most important functions currently studied today.

  Finally, Ramanujan formulated a conjecture about the rate of growth of the tau function. For every prime number p, he conjectured that −2p 11/2 < τ(p) < 2p 11/2. He thus conjectured that each of these numbers is restricted to a certain range. It shouldn’t be too large or too small, and the conjecture makes precise what is meant by “large” and “small.” This conjecture was further developed and folded into deep conjectures in algebraic geometry formulated by Weil, which were then proved by Deligne, earning him a Fields Medal.

  Ramanujan invented his function tau in his 1916 paper, and the results he obtained seemed to be nothing more than oddities: strange-looking formulas and conjectured inequalities. Before him, nobody would have cared about this mathematics. It was his genius that recognized the value of these ideas, and it was up to mathematicians of the future to recognize their importance and make use of them. And so they did, and from the seeds that Ramanujan planted, a magnificent garden has grown.

  Ken Ono and Dev Patel rehearsing the “Circle Method Scene” (photo by Sam Pressman)