Ramanujans intuited connection between two mathematical functions proved

Updated on: Saturday, November 10, 2012

A US scientist has finally proven the mathematical relation — discussed by great Indian mathematician Srinivasa Ramanujan with his mentor G.H. Hardy in one of his last letters.

Mr. Ramanujan, the young genius who died at the age of 32, often managed to leap from insight to insight without formally proving the logical steps in between.

Now proof has been found for a connection that he seemed to mysteriously intuit between two types of mathematical functions, the New Scientist reported.

G.H. Hardy, Mr. Ramanujan’s mentor and one of his few collaborators, had once said, “His ideas as to what constituted a mathematical proof were of the most shadowy description.”

Ken Ono of Emory University in Atlanta, Georgia, who has previously unearthed hidden depths in Mr. Ramanujan’s work, was prompted by Mr. Ramanujan’s 125th birth anniversary, to look once more at his writings.

“I wanted to go back and prove something special,” Mr. Ono said.

Ono settled on a discussion in the last known letter that Mr. Ramanujan wrote to Mr. Hardy, concerning a type of function now called 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

The functions looked unlike any other modular forms, but Mr. Ramanujan wrote that their outputs would be very similar to those of modular forms when computed for the roots of 1 like the square root —1.

Characteristically, Mr. Ramanujan offered neither proof nor explanation for this conclusion.

It was only 10 years later that mathematicians formally defined this other set of functions, now known as mock modular forms.

However, still no one fathomed what Mr. Ramanujan meant by saying the two types of function produced similar outputs for roots of 1.

Now Mr. Ono and colleagues have exactly calculated one of Mr. Ramanujan’s mock modular forms for values very close to —1, and said the difference in the value of the two functions, ignoring the functions signs, is tiny when computed for —1, just like Mr. Ramanujan said.

The result confirms Mr. Ramanujan’s incredible intuition, said Mr. Ono.

Ramanujans intuited connection between two mathematical functions proved

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