MOCK THETA FUNCTIONS. Is the secret to Ramanujan's quip about the number 1729 in the following formula?

Milwaukee Journal-Sentinel image (using Maple?)

Sometimes the secret comes in making connections.

Ken Ono was reading an article by George Andrews in which the Pennsylvania State University professor listed six great problems for mathematicians to solve in the new millennium. The last two problems referred to the baffling functions first described in 1920 by Indian mathematician Srinivasa Ramanujan on his deathbed.

As with much of the work in his short life - Ramanujan died at 32 - he set down the mock theta functions without showing how he knew they were true, without even showing the trail of thought that led to his discoveries. For decades, mathematicians have regarded the functions as tantalizing clues - but to what?

Now, as Ono read the article, he realized that he knew how to solve one of the problems. Some of the formulas needed were similar to those he and colleague Kathrin Bringmann had been using for a theory they were developing.

He rushed down the aisle to Bringmann's seat.

"Kathrin," Ono said, "you have to read this. They look exactly like our functions."

So far, that high-altitude flash of insight has led to three papers by Ono, 38, and Bringmann, 29. In the first, they solved Andrews' fifth problem for the millennium. In the remaining two, they solved Problem 6, "more or less," Ono said.

Professor Andrews may be too busy to create his own web page. This sketch provides some information about his work. He had a good start.

Ironically, it was outside his mathematics courses where Andrews was first exposed to partition functions. His then-fiancée, Joy, now his wife of 44 years, gave him a four-volume set of books called The World of Mathematics, which contained Godfrey H. Hardy's A Mathematician's Apology (2). Buried in a footnote was a comment about a surprising mathematical formula that Hardy and Ramanujan had uncovered together in 1916. The elegance of the formula struck Andrews. "I thought it was just stunning," he says. This fundamental formula in partition theory expresses the number of ways an integer can be broken down into natural number summands. For example, there are three partitions of the number 3 (3, 2+1, and 1+1+1) and five partitions of the number 4 (4, 3+1, 2+2, 2+1+1, and 1+1+1+1). What Hardy and Ramanujan had found was an exact formula for the number of partitions of an integer. "It doesn't seem like you'd need an exact formula," Andrews explains, "but while there are only five partitions of 4, there are almost 4 trillion partitions of 200." Even computers could not handle this task as well as an exact formula could, he says, so this result was as useful as it was elegant.

There's more to his discovery of Ramanujan's lost notebook than sheer serendipity.

He took advantage of the trip to Europe to visit Trinity College (Cambridge, U.K.), where he browsed through papers from the estate of the late mathematician George N. Watson, which were housed there.

What Andrews found among the dusty papers grabbed his attention. In one box lay about 100 loose pages filled with, of all things, equations in Ramanujan's handwriting. And there, Andrews realized a few minutes later, were more of the legendary mock theta functions that Ramanujan had hinted at. "It was a gold mine," Andrews says.

Merely recognizing Ramanujan's handwriting was not the key to Andrews' discovery, his colleague Askey explains. The real detective feat was spotting, in row after row of unlabeled formulas, equations fitting the bill of those enigmatic functions Ramanujan had described. "There were certain identities that George [Andrews] recognized as mock theta functions," Askey says. "No one else would have spotted them instantly. George has done many things, but this is what will make the history books."

(And thus my gripe with "Social Justice Lego." Resources that happen to be lying around are of no value unless somebody recognizes the value. Many others might have leafed through the Ramanujan notebook without understanding what they were seeing.)

As the Journal-Sentinel report puts it,

In a 1920 letter to his British mentor and collaborator G.H. Hardy, Ramanujan said he had discovered a new class of theta functions and set down 17 examples, the mock theta functions. More than 50 years later, a half-dozen or so additional examples surfaced when Andrews, then a visiting professor at UW, stumbled upon 140 pages of previously undiscovered Ramanujan work - the so-called "Lost Notebook" - in the library of Trinity College at the University of Cambridge. Until now, so little was known about these functions that mathematicians have yet to explore all of their uses.

Professor Ono (also a Ramanujan scholar, in his spare time working on modular forms) and Professor Bringman (modular forms and partition functions) have made the paper available as .pdf. (As is the case with many marvelous proofs, this one is too large to fit in the margin.)