杏吧原创

PROOFS are the currency of mathematics, but Srinivasa Ramanujan, one of the all-time great mathematicians, often managed to skip them. Now a proof has been found for a connection that he seemed to mysteriously intuit between two types of mathematical function.

The proof deepens the intrigue surrounding the workings of Ramanujan鈥檚 enigmatic mind. It may also help physicists learn more about black holes 鈥� even though these objects were virtually unknown during the Indian mathematician鈥檚 lifetime.

Born in 1887 in Erode, Tamil Nadu, Ramanujan was self-taught and worked in almost complete isolation from the mathematical community of his time. Described as a raw genius, he independently rediscovered many existing results, as well as making his own unique contributions, believing his inspiration came from the Hindu goddess Namagiri. But he is also known for his unusual style, often leaping from insight to insight without formally proving the logical steps in between. 鈥淗is ideas as to what constituted a mathematical proof were of the most shadowy description,鈥� said G. H.Hardy (pictured, far right), Ramanujan鈥檚 mentor and one of his few collaborators.

聯His ideas as to what constituted a mathematical proof were of the most shadowy description聰

Despite these eccentricities, Ramanujan鈥檚 work has often proved prescient. This year is the 125th anniversary of his birth, prompting Ken Ono of Emory University in Atlanta, Georgia, who has previously unearthed hidden depths in Ramanujan鈥檚 work, to look once more at his notebooks and letters. 鈥淚 wanted to go back and prove something special,鈥� says Ono. He settled on a discussion in the last known letter penned by Ramanujan, to Hardy, concerning a type of function now known as a modular form.

Functions are equations that can be drawn as graphs on an axis, like a sine wave, and produce an output when computed for any chosen input or value. In the letter, Ramanujan wrote down a handful of what were then totally novel functions. They looked unlike any known modular forms, but he stated that their outputs would be very similar to those of modular forms when computed for the roots of 1, such as the square root -1. Characteristically, Ramanujan offered neither proof nor explanation for this conclusion.

It was only 10 years ago that mathematicians formally defined this other set of functions, now called mock modular forms. But still no one fathomed what Ramanujan meant by saying the two types of function produced similar outputs for roots of 1.

Now Ono and colleagues have exactly computed one of Ramanujan鈥檚 mock modular forms for values very close to -1. They discovered that the outputs rapidly balloon to vast, 100-digit negative numbers, while the corresponding modular form balloons in the positive direction.

Ono鈥檚 team found that if you add the corresponding outputs together, the total approaches 4, a relatively small number. In other words, the difference in the value of the two functions, ignoring their signs, is tiny when computed for -1, just as Ramanujan said.

The result confirms Ramanujan鈥檚 incredible intuition, says Ono. While Ramanujan was able to calculate the value of modular forms, there is no way he could have done the same for mock modular forms, as Ono now has. 鈥淚 calculated these using a theorem I proved in 2006,鈥� says Ono, who presented his insight at the in Gainesville, Florida, this week. 鈥淚t is inconceivable he had this intuition, but he must have.鈥�

Figuring out the value of a modular form as it balloons is comparable to spending a coin in a particular shop and then predicting which town that coin will end up in after a year.

Guessing the difference between regular and mock modular forms is even more incredible, says Ono, like spending two coins in the same shop and then predicting they will be very close a year later.

Though Ono and colleagues have now constructed a formula to calculate the exact difference between the two types of modular form for roots of 1, Ramanujan could not possibly have known the formula, which arises from a bedrock of modern mathematics built after his death.

鈥淗e had some sort of magic tricks that we don鈥檛 understand,鈥� says of the Institute for Advanced Study in Princeton, New Jersey.

While modular forms are mostly related to abstract problems, Ono鈥檚 formula could have applications in calculating the entropy of black holes (see 鈥�The black hole connection鈥�).

So will Ono鈥檚 work turn out to be the last of Ramanujan鈥檚 contributions? 鈥淚鈥檓 so tempted to say that,鈥� says Ono. 鈥淏ut I won鈥檛 be surprised if I鈥檓 dead wrong.鈥�