Mock Theta Functions: Ideas Ahead of His Time

Mathematical concepts that took 80 years to fully understand

Discover Ramanujan's mock theta functions, mysterious mathematical objects whose true nature was not understood until the twenty-first century.

Mock Theta Functions: Ideas Ahead of His Time

On January 12, 1920, a twenty-eight-year-old, very sick man in a small house in Kumbakonam dictated one last letter to his collaborator G. H. Hardy in Cambridge. He had come home from England ten months earlier. Tuberculosis, or possibly hepatic amoebiasis, had reduced him to skin over bone. He would die in three months. The letter, barely ten handwritten pages, contained no complaints, no farewells, and only the briefest gossip. What it contained instead was a new class of mathematical object that he had invented and named himself: mock theta functions. He listed seventeen of them, arranged by 'order' into groups of three, five, and seven, and offered almost no explanation of what the words 'mock theta' were supposed to mean.

For eighty-two years, that letter was one of the great unsolved riddles of modern mathematics.

A Letter Without a Theory

Ramanujan was accustomed to writing down results that other mathematicians did not yet know how to reach. He had done it in his Indian notebooks, in his first letter to Hardy in 1913, and throughout his Cambridge years. By the standards of his own earlier output, the January 1920 letter was unusual for a different reason. It did not just state results. It introduced a name for a kind of function, gave examples, hinted at properties, and then stopped. The letter said, in effect, that these things exist, they are important, they are not theta functions even though they look like theta functions from some angles, and here are seventeen of them. Go figure out why.

Hardy filed the letter, spoke of it after Ramanujan's death as part of the 'final' work, and helped ensure its contents were published. For most of the twentieth century, the mock theta functions sat in the literature as an exotic collection of formulas that clearly pointed at something and just as clearly were not yet understood. Number theorists played with them. Physicists noticed them. Nobody could say what they really were.