An uncanny coincidence recently led two University of Wisconsin number theorists to solve a puzzle posed more than 85 years ago by self-taught Indian mathematician Srinivasa Ramanujan (1887-1920).
The problem concerns mock theta functions. Shortly before his death in 1920, Ramanujan wrote a last letter to his collaborator, G.H. Hardy (1877#&150;1947). In his letter, Ramanujan introduced 17 new functions. He asserted that these functions shared certain properties with theta functions, which had been investigated at great length by Carl Gustav Jacob Jacobi (1804-1851). Theta functions are essentially elliptic analogs of the exponential function. Jacobi proved a variety of identities and found expressions for these special functions in terms of infinite series and infinite products.
However, Ramanujan's letter provided no hint of where his "mock" theta functions come from, how to derive them, and why they might be important or interesting.
In the 1980s, George Andrews and F.G. Garvan conjectured that two families of Ramanujan's theta functions are genuinely different from Jacobi's theta functions, not just the same functions in disguise. They linked Ramanujan's functions to partitions of a given integer—the ways of writing an integer as a sum of smaller integers. Building on the work by Andrews and Garvan, Dean Hickerson proved that five identities in each of the two families are equivalent, at the same time confirming that these mock theta functions are truly mock theta functions.
But, despite the efforts of many mathematicians over the years, the functions themselves remained mysterious. In 1987, Freeman Dyson noted, "The mock theta functions give us tantalizing hints of a grand synthesis still to be discovered . . . . This remains a challenge for the future. "
Now, Ken Ono and Kathrin Bringmann have constructed an explanatory framework that, for the first time, shows what mock theta functions are and how to derive them. They link Ramanujan's functions to mathematical objects known as modular forms, specifically to a special class of mathematical expressions known as Maass forms. Modular forms have recently played important roles in number theory, including the proof of Fermat's last theorem.
Mathematician Sander P. Zwegers had made the connection between Ramanujan's functions and Maass forms in 2002, but he had focused on only a handful of examples. Ono and Bringmann had been working on Maass forms for other reasons. Then, while on a flight, Ono happened upon an old article by George Andrews on mock theta functions. He was startled to find that some of the mathematics in that paper echoed parts of Maass theory. Upon investigating the connection in greater detail with Bringmann, Ono found that it held up, providing a coherent way of understanding what mock theta functions really represent.
The link that Bringmann and Ono have established fits a grand vision in number theory—one that Ono describes as the “web of modularity.” The idea is that most of the entities of interest in number theory are in some sense controlled or parametrized by modular forms or their generalizations.
At the same time, another link has now been forged between the remarkably prescient work of Ramanujan and the cutting edge of research in number theory.—I. Peterson
2007. UW scientists unlock major number theory puzzle. University of Wisconsin press release. Feb. 26. Article.
Bringmann, K., and K. Ono. In press. Arithmetic properties of coefficients of half-integral weight Maass-Poincaré series. Mathematische Annalen.
______. 2007. Lifting cusp forms to Maass forms with an application to partitions. Proceedings of the National Academy of Sciences 104(March 6):3725-3731. Abstract.
______. 2006. The f(q) mock theta function conjecture and partition ranks. Inventiones Mathematicae 165:243-266. Article.
Ono, K. Preprint. Mock theta functions, ranks, and Maass forms. Article.