The Master of the Unseen

Srinivasa Ramanujan and the Mock Theta Partition

The eleven-page packet that arrived on G. H. Hardy's breakfast table at Trinity College, Cambridge, on the morning of January 31, 1913, bore cheap Indian postage stamps and contained not a single formal proof. Postmarked from Madras and signed by shipping clerk at the Madras Port Trust earning twenty rupees a month—roughly twenty British pounds a year—the cover letter begged the famous English analyst to examine an enclosed list of 120 theorems on infinite series, integrals, and prime numbers.

Hardy, the most rigorous pure mathematician in Britain, initially glanced at the pages over morning coffee and suspected an elaborate fraud. The manuscript included bizarre assertions such as $1 + 2 + 3 + 4 + \dots = -\frac{1}{12}$, alongside sprawling continued fractions evaluated to expressions involving $\pi$ and $e$. Yet after dinner that evening, when Hardy and his collaborator John Edensor Littlewood sat down in Trinity College to dissect the formulas line by line for three hours, their skepticism evaporated. A third of the theorems were known nineteenth-century results by Carl Gustav Jacobi and Bernhard Riemann that the isolated clerk had somehow rediscovered from scratch; another third were new and deeper than anything in European journals; and the final group of identities defeated both Cambridge professors completely. As Hardy later remarked, those last formulas had to be true, because no human being possessed the imagination to invent them falsely.

Srinivasa Ramanujan had grown up in the temple town of Kumbakonam in southern India. At age fifteen, a local college student had lent him an obscure 1886 English coaching manual: George Shoobridge Carr's A Synopsis of Elementary Results in Pure and Applied Mathematics. Designed to drill Cambridge examination candidates, Carr's book listed 4,865 mathematical theorems in terse summary form without full derivations. Because Ramanujan became so obsessed with mathematics that he failed his English, physiology, and Sanskrit exams at the Government Arts College in Kumbakonam and Pachaiyappa's College in Madras—losing his scholarship twice—he worked for years in extreme poverty on the front porch of his family home. Because paper was expensive, he worked out his derivations with chalk on a handheld slate, wiping the slate clean with his elbow and copying only the final compressed formulas into three leather-bound notebooks.

When Hardy brought Ramanujan to Cambridge in April 1914, their collaboration joined two opposite cognitive styles: Ramanujan's direct perception of algebraic symmetries and Hardy's command of rigorous complex analysis. Together they attacked the classical problem of integer partitions—determining $p(n)$, the number of distinct ways a whole number $n$ can be written as a sum of positive integers. For $n = 4$, there are five partitions (4, $3+1$, $2+2$, $2+1+1$, and $1+1+1+1$), so $p(4) = 5$. Yet $p(n)$ grows at an explosive combinatorial rate; for $n = 200$, Major Percy MacMahon had spent weeks of manual calculation showing that $p(200) = 3,972,999,029,388$. Every previous attempt to find an exact formula had failed because $p(n)$ is a discrete step function controlled by prime divisibilities.

In 1918, Hardy and Ramanujan invented the analytic circle method. Treating the generating function of $p(n)$ as a complex function inside the unit disk, they noticed that the function exploded not just at $z = 1$, but at every root of unity $e^{2\pi i h / k}$ around the boundary circle. By integrating along a circular contour and summing the contributions of all fractional roots of unity, they derived an exact convergent series whose leading asymptotic term is:

Using only the first five terms of their full series, Ramanujan computed $p(200)$ to within 0.004 of MacMahon's thirteen-digit integer. Stricken with severe hepatic amoebiasis worsened by wartime rationing in England, Ramanujan returned to India in 1919 and died on April 26, 1920, at age thirty-two. In 2002, Sander Zwegers finally proved that the seventeen mysterious 'mock theta' series Ramanujan scribbled on his deathbed in 1920 are the holomorphic parts of harmonic Maass forms, a mathematical structure physicists now use to compute the quantum entropy of black holes.