Ramanujan’s Divine Mathematics

Researchers recently solved the cryptic deathbed puzzle renowned Indian mathematician Srinivasa Ramanujan claimed came to him in dreams. While on his death-bed in 1920, Ramanujan wrote a letter to his mentor, English mathematician G. H. Hardy, outlining several new mathematical functions never before heard of, along with a hunch about how they worked, decades later, researchers say they’ve proved he was right – and that the formula could explain the behaviour of black holes.

For people who work in this area of math, the problem has been open for almost 90 years. Ramanujan’s letter described several new functions that behaved differently from known theta functions, or modular forms, and yet closely mimicked them.

 

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. Ramanujan conjectured that his mock modular forms corresponded to the ordinary modular forms earlier identified by Carl Jacobi, and that both would wind up with similar outputs for roots of 1. Ramanujan prime and theta functions, have inspired vast amounts of further research and have have found applications in fields as diverse as crystallography and string theory.

Ramanujan, a devout Hindu, thought these patterns were revealed to him by the goddess Namagiri. December 22, 2012 marks the 125th birth anniversary of Srinivasa Ramanujan, the self taught mathematician born into a modest and conservative family in Kumbakonam, a relatively small town in Tamilnadu.

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. He spent so much time thinking about math that he flunked out of college in India twice. He overcame several hurdles to find a place among the celebrated intellectuals of Cambridge. Ramanujan passed away at the young age of 32 of tuberculosis, but he left behind formulations in mathematics that have paved the path for many scholars who came after him.

It is estimated that Ramanujan conjectured or proved over 3,000 theorems, identities and equations, including properties of highly composite numbers, the partition function and its asymptotics and mock theta functions. He also carried out major investigations in the areas of gamma functions, modular forms, divergent series, hypergeometric series and prime number theory.

Among his other achievements, Ramanujan identified several efficient and rapidly converging infinite series for the calculation of the value of π, some of which could compute 8 additional decimal places of π with each term in the series. These series (and variations on them) have become the basis for the fastest algorithms used by modern computers to compute π to ever increasing levels of accuracy (currently to about 5 trillion decimal places).

A common anecdote about Ramanujan during this time relates how Hardy arrived at Ramanujan’s house in a cab numbered 1729, a number he claimed to be totally uninteresting. Ramanujan is said to have stated on the spot that, on the contrary, it was actually a very interesting number mathematically, being the smallest number representable in two different ways as a sum of two cubes. Such numbers are now sometimes referred to as “taxicab numbers”.

* Ramanujan’s note books can be found here