# Ramanujan and Theta Functions

While on his death bed, the brilliant Indian mathematician savant Srinivasa Ramanujan wrote down cryptic mathematical functions he said came to him in dreams from the Hindu goddess Namagiri. Now, amazingly, 90 years later, researchers say they’ve proved he was right.

Ramanujan, was a brilliant self-taught mathematician born in a poor rural village in Southern India, had a natural talent for mathematics from a very early age. He made such contributions to mathematics in such a short time that, in recognition the Government of India has designated his birthday (22 December) as the annual National Mathematics Day.

A Genius in Isolation By the time he was 11, he had already surpassed the mathematical knowledge of two college students who were lodgers at his home. He was then lent a textbook on advanced trigonometry written by S. L. Loney which truly sparked his intellectual fervour. At 14 his intelligence and capability was recognised to the point that he was entrusted with organising the logistics of his own school.

Living in India with no access to the larger mathematical community, which was very much focused in Europe at the time, Ramanujan had to develop his own mathematical research with little outside input. Therefore he sometimes rediscovered known theorems as well as producing new work which hugely impressed his peers. Ramanujan was recognized as a natural genius by the English mathematician G. H. Hardy who took a great interest in his career.

Life in the UK Hardy brought him to the UK and took him under his wing, attempting to re-educate him in formal mathematical theory.. Ramanujan’s genius was often frustratingly intuitive and obscure, as he tended to made huge leaps in mathematical reasoning without demonstrating the logical progression of thought and proofs.

Sadly the cold weather, European germs and pure exhaustion from overwork and isolation meant that Ramanujan died at the tragically young age of 32. “For a brief window of time, five years, he lit the world of math on fire,” says Emory University mathematician Ken Ono, but his unique brilliance was destined to blaze brightly but ultimately consume him. He went home to India to die but continued to write to Hardy and pass on his increasingly erratic but fascinating mathematical insights.

Mysterious Functions It was on his deathbed that he described the mysterious functions that mimicked theta functions, or modular forms, in a letter to Hardy. Like trigonometric functions such as sine and cosine, theta functions have a repeating pattern, but the pattern is more complicated than a sine curve.

Theta functions are also “super-symmetric” meaning that if a specific type of mathematical function called a Moebius transformation is applied to the functions, they turn into themselves. Because they are so symmetric, these theta functions are useful in many types of mathematics and physics.

Ramanujan died before he could prove these theories but finally more than 90 years later, Ono and his team have proved that these functions do indeed actually mimic modular forms.

Ono says. “We found the formula explaining one of the visions that he believed came from his goddess.” Ramanujan could not possibly have known this formula, which arises from a bed-rock of modern mathematics built after his death. Ono said. “It is inconceivable he had this intuition, but he must have.” The proof deepens further the intrigue surrounding the workings of Ramanujan’s enigmatic mind.

Unlocking the Secrets of the Stars The team were also astonished to realize the functions are useful today when studying advanced ideas barely imagined when Ramanujan was alive, such as String Theory and the study of Black Holes. “No one was talking about black holes back in the 1920s when Ramanujan first came up with mock modular forms, and yet, his work may unlock secrets about them,” Ono says.

At ConquerMaths.com we think stories like this show the incredible potential for exciting discovery in the world of maths and it shows no matter what your background you have a chance to engage with this world in a significant way. We may never fully understand how this poor young Indian student made these amazing leaps in mathematical reasoning but generations to come will benefit from his insights and we will surely discover even more insights into his fascinating genius for years to come.