# Ramanujan: from his last letter to the lost notebook

Reaching out: In his last letter to Hardy in 1920, Ramanujan wrote, I discovered very interesting functions recently, which I call ‘mock’ theta-functions.   | Photo Credit: The Hindu

In honour of “magical” mathematician, Srinivasa Ramanujan who died exactly a hundred years ago, in 1920, Vigyan Prasar organised events under the name “Ramanujan Yatra” which included a series of talks held monthly on Ramanujan’s work.

“Ramanujan’s life carries the message that perseverance leads to results, and that one should not give up in the face of adversity,” says T.V. Venkateswaran of Vigyan Prasar.

## Chance discovery

In 1976, almost entirely by chance, George E. Andrews, who is Evan Pugh professor of mathematics, Pennsylvania State University, found Ramanujan’s notes written during his last few years in England. Prof. Andrews, along with Bruce C. Berndt went on to compile the contents of this lost notebook into a five-volume book entitled Ramanujan’s Lost Notebook. This book, which is a mathematical treasure trove, took them twenty years to bring out, such was its complexity. Prof. Andrews gave a lecture in the Ramanujan Yatra wherein he described how he discovered the lost notebook – and the gems embedded in it.

“We wanted to invite noted Ramanujan scholars from all over the world, with the idea that undergraduate students would be inspired by Ramanujan’s mathematics, and hopefully study more on their own,” says R. Ramanujam, computer scientist from The Institute of Mathematical Sciences, Chennai, who co-organised the lecture series.

## Last letter

In Ramanujan’s last letter to G H Hardy, dated January 12, 1920, he had written, “I discovered very interesting functions recently, which I call ‘mock theta-functions’. Unlike the ‘false’ theta-function (studied partially by Prof Rogers in his interesting paper) they enter into mathematics as beautifully as the ordinary theta-function.” He proceeds to give some examples and relationships. He proceeds to give some examples and relationships, and in the last page of the letter, lists three functions, calling them third order, fifth order and seventh order and ends with a cryptic statement – “these functions are not related to one another.”

Ramanujan died on April 26, 1920. As Prof. Andrews observed, the notes Ramanujan had made when still in England around the years 1918-1919 may have been handed over by his mentor G.H. Hardy to mathematician G.N. Watson for editing and compiling.

In 1965, Prof. Watson died, and in 1976, when Prof Andrews visited the family, he was invited to look at the collection that the former had left behind and to take his pick. The collection “covered the floor of a fair-sized room to the depth of a foot and were to be incinerated in a few days,” observed Prof. Andrews. It consisted of mathematical documents, but also receipts, tokens etc. Prof Watson was an avid collector.

With his “lucky dip” Prof. Andrews came up with a set of notes. The equations there were written without any introduction or explanation – just numbers and bare equations. As luck would have it, Prof. Andrews had studied the mock theta functions earlier and recognised them despite the unconventional notation. Having read Ramanujan’s last letter, he recognised this was the latter’s last work. He held in his hand Ramanujan’s lost notebook!

The pages were full of equation after equation, some related, many not related, some pages in organised form and nearly 40% were chaotic, according to Prof Andrews, who also narrated an interesting story of how Ramanujan had worked in advance on conjectures made later by other mathematicians, such as Freeman Dyson.

## Dyson’s conjectures

In 1944, Freeman Dyson had made an extremely unusual conjecture related to Ramanujan’s partition congruences. This was called the “crank” of the partition function. Australian-born mathematician Frank Garvan, presently at University of Florida, who was Prof. Andrews’s PhD student resolved this problem in his PhD thesis.

As Prof. Andrews noted, “Frank Garvan’s thesis is devoted to an expanded study of this page [from the Lost Notebook] and proof of the identities.” Garvan showed that a later proof of Dyson’s conjecture by Atkin and Swinnerton Dyer was equivalent to what Ramanujan had noted. He also revealed that the function Ramanujan had noted down was indeed the “crank” famously predicted by Dyson

## Anticipating maths

Thus, according to Prof. Andrews, “Ramanujan not only anticipated Dyson’s conjecture, but also anticipated his conjecture about conjectures.” Prof. Andrews in his talk went on to highlight several other gems from the Lost Notebook that reiterated that Ramanujan had anticipated the mathematics that came later. On a personal note, Prof. Ramanujam says, “The fellowship Ramanujan received in Cambridge was meagre, but he had so few needs that he saved half of the money and sent it home, with instructions that a substantial part of it be used to support indigent students of Madras university.”