Rediscovering Ramanujan

Print edition : August 14, 1999

The academic lineage of most eminent scholars can be traced to famous centres of learning, inspiring teachers or an intellectual milieu, but Srinivasa Ramanujan, perhaps the greatest of Indian mathematicians, had none of these advantages. He had just one year of education in a small college; he was basically self-taught. Working in isolation for most of his short life of 32 years, he had little contact with other mathematicians.

"Many people falsely promulgate mystical powers to Ramanujan's mathematical thinking. It is not true. He has meticulously recorded every result in his three notebooks," says Dr. Bruce C. Berndt, Professor of Mathematics at the University of Illino is, whose 20 years of research on the three notebooks has been compiled into five volumes.

Between 1903 and 1914, before Ramanujan went to Cambridge, he compiled 3,542 theorems in the notebooks. Most of the time Ramanujan provided only the results and not the proof. Berndt says: "This is perhaps because for him paper was unaffordable and so he worked on a slate and recorded the results in his notebooks without the proofs, and not because he got the results in a flash."


Berndt is the only person who has proved each of the 3,542 theorems. He is convinced that nothing "came to" Ramanujan but every step was thought or worked out and could in all probability be found in the notebooks. Berndt recalls Ramanujan's well-known i nteraction with G.H. Hardy. Visiting Ramanujan in a Cambridge hospital where he was being treated for tuberculosis, Hardy said: "I rode here today in a taxicab whose number was 1729. This is a dull number." Ramanujan replied: "No, it is a very interestin g number; it is the smallest number expressible as a sum of two cubes in two different ways." Berndt believes that this was no flash of insight, as is commonly thought. He says that Ramanujan had recorded this result in one of his notebooks before he cam e to Cambridge. He says that this instance demonstrated Ramanujan's love for numbers and their properties.

Although Ramanujan's mathematics may seem archaic by today's standards, in many respects he was far ahead of his time. While the thrust of 20th century mathematics has been on building general theories, Ramanujan was a master in finding particular result s which are now recognised as providing the core for the theories. His results opened up vistas for further research not only in mathematics but in other disciplines such as physics, computer science and statistics.

After Ramanujan's death in 1920, the three notebooks and a sheaf of papers that he left behind were handed over to the University of Madras. They were sent to G.N. Watson who, along with B.M. Wilson, edited sections of the notebooks. After Watson's death in 1965, the papers, which contained results compiled by Ramanujan after his return to India from Cambridge in 1914, were handed over to Trinity College, Cambridge. In 1976, G. E. Andrews of Pennsylvania State University rediscovered the papers at the T rinity College Library. Since then these papers have been called Ramanujan's "lost notebook". According to Berndt, the lost notebook caused as much stir in the mathematical world as Beethoven's Tenth Symphony did in the world of Western classical music.

Berndt says that the "unique circumstances surrounding Ramanujan and his mathematics" make it very difficult to assess his greatness among such mathematical giants as Newton, Gauss, Euler and Reimann. According to Berndt, Hardy had provided the following assessment of his contemporary mathematicians on a scale of 0 to 100: "On the basis of pure talent he gave himself a rating of 25, his collaborator J. E. Littlewood 30, German mathematician D. Hilbert 80, and Ramanujan 100." Berndt says that it is not R amanujan's greatness but only its measure that is in doubt.

Besides the five volumes, Berndt has written over 100 papers on Ramanujan's works. He has guided a number of research students in this area. He now works on Ramanujan's "lost notebook" and on some other manuscripts and fragments of notes. Recently in Che nnai to give lectures on Ramanujan's works at the Indian Institute of Technology, the Institute of Mathematical Sciences and the Ramanujan Museum and Mathematical Centre, Berndt spoke to Asha Krishnakumar on his work on Ramanujan's notebooks, the broad areas in mathematics that Ramanujan had covered, the vistas his work has opened up and the application of his work in physics, statistics and communication.

Excerpts from the interview:

How did you get interested in Ramanujan's notebooks?

After my Ph.D. at the University of Wisconsin, I took my first position at the University of Glasgow (Scotland) in 1966-67. Prof. R. A. Rankin was a leader in number theory at that time. I remember being in Rankin's office in 1967 when he told me about R amanujan's notebooks for the first time. He said: "I have a copy of the notebooks published by the Tata Institute of Fundamental Research, Bombay (Mumbai). Would you be interested in looking at it?" I said, "No, I am not interested in it."

I did not think about the notebooks for some years until early 1974 when I was on leave at the Institute for Advanced Studies in Princeton, U.S. In February that year, I was reading two papers of Emil Grosswald in which he proves some formulae from Raman ujan's notebooks. I realised I could prove these formulae as well by using a theorem I proved two years ago. I did that and then I was curious to find out whether there were other formulae in the notebooks that I could prove using my methods. So, I went to the Princeton University library and got hold of Ramanujan's notebooks published by the TIFR. I was thrilled to find out that I could actually prove some more formulae. But there were a few thousand others I could not.

I was fascinated with the notebooks and in the next few years I wrote papers around the formulae I had proved from the notebooks. The first was a repository paper on Ramanujan's theta 2n+1 formula, for which I did a lot of historical research on other pr oofs of the formula. This I wrote for a special volume called Srinivasa Ramanujan's Memorial Volume, published by Jupiter Press in Madras (Chennai) in 1974. After that, wherever I went, I was all the time working on, and proving, the various formulae of Ramanujan's - to be precise - from Chapter 14 of the second notebook. Then I wrote a sequel to this.

Let me jump ahead to May 1977, when I decided to try and prove all the formulae in Chapter 14. I took this on as a challenge. There were in all 87 results in this chapter. I worked on this for the next one year. I took the help of my first Ph.D. student, Ron Evans.

After about a year of working on this, the famous mathematician George E. Andrews visited Illinois and told me that he discovered in the spring of 1976 Ramanujan's "lost notebook" along with G. N. Watson and B. M. Wilson's edited volumes on Ramanujan's t hree notebooks and some of their unpublished notes in the Trinity College Library. I then got photocopies of Ramanujan's lost notebook and all the notes of Watson and Wilson. And so I went to the beginning of the second notebook.

What does the second notebook contain?

This is the main notebook because it is the revised and enlarged version of the first. I went back to the beginning and went about working my way through it using Watson and Wilson's notes when necessary.

How long did you work on the second notebook?

I really do not know how many years exactly. But some time in the early 1980s Walter Kaufmann-Buhler, the mathematics editor of Springer Verlag in New York, showed interest in my work and decided to publish it. That had not occurred to me till the n. I agreed and signed a contract with Springers.

That was when I started preparing the results with a view to publishing them. I finally came out with five volumes; I had thought it would be three. It also took a much longer time than I had anticipated.

After I completed 21 chapters of the second notebook, the 100 pages of unorganised material in the second notebook and the 33 pages in the third had a lot more material. I also found more material in the first which was not there in the second. So, I fou nd a lot of new material. It was 20 years before I eventually completed all the three notebooks.

Why did you start with the second notebook and not the first?

I knew that the second was the revised and enlarged edition of the first. The first was in a rough form and the second, I was relatively certain, had most of the things that were there in the first and a lot more.

What did each notebook contain?

The new results that were in the second notebook were generally among the unorganised pages of the first. And the third notebook was all unorganised. A higher percentage of the results in the unorganised parts of the second and the third were new. In oth er words, you got a higher percentage of new results as you went into the unorganised material.

What do you mean by new results? Results that have not been got earlier.

What is the percentage of new results in the notebooks?

Hardy estimated that over two-thirds of the work Ramanujan did in India was rediscovered. That is much too high. I found that well over half is new. It is difficult to say precisely. I would say that most results were new because we also have to consider that in the meantime, from 1920 until I started doing this work, other people discovered these things. So, I would say that at least two-thirds of the material was really new when Ramanujan died.

Srinivasa Ramanujam.-THE HINDU PHOTO LIBRARY

Ramanujan is popularly known as a number theorist. Would you give a broad idea about the results in his notebooks? What areas of mathematics do they cover?

You are right. To much of the mathematical world and to the public in general, Ramanujan is known as a number theorist. Hardy was a number theorist but he was also into analysis. When Ramanujan was at Cambridge with Hardy, he was naturally influenced by him (Hardy). And so most of the papers he published while he was in England were in number theory. His real great discoveries are in partition functions.

Along with Hardy, he found a new area in mathematics called probabilistic number theory, which is still expanding. Ramanujan also wrote sequels in highly composite numbers and arithmetical functions. There are half a dozen or more of these papers that ma de Ramanujan very famous. They are still very important papers in number theory.

However, the notebooks do not contain much of number theory. It is, broadly speaking, in analysis. I will try and break that down a little bit. I would say that the area in which Ramanujan spent most of his time, more than any other, is in elliptic funct ions (theta functions), which have strong connections with number theory. In particular, Chapters 16 to 21 of the second notebook and most of the unorganised portions of the notebooks are on theta functions. There is a certain type of theta functions ide ntity which has applications in other areas of mathematics, particularly in number theory, called modular equations. Ramanujan devoted an enormous amount of effort on refining modular equations.

Ramanujan is also popular for his approximations to pie. Many of his approximations came with his work on elliptic functions. Ramanujan computed what are called class invariants. Even as he discovered them, they were computed by a German mathematician, H . Weber, in the late 19th and early 20th centuries. But Ramanujan was unaware of this. He computed 116 of these invariants which are much more complicated. These have applications not only in approximations to pie but in many other areas as well.

Have you gone through every one of the 3,254 entries in the three notebooks and proved each of them, including in the unorganised material?

I have gone through every entry in the notebooks. If a result has already been proved in the literature, then I just wrote the entry down and said that proofs can be found in this literature and so on. But I will also discuss the relevance in history of the entry.

What are the applications of Ramanujan's discoveries in areas such as physics, communications and computer science?

This is a very difficult question to answer because of the way mathematics and science work. Mathematics is discovered and it is then there for others to use. And you do not always know who uses it. But I have regular contact with some physicists who I k now use Ramanujan's work. They find the results very useful in their own application.

What are the areas in physics in which Ramanujan's work is used?

The most famous application in physics is in the area of statistical mechanics. Among those who I know have used Ramanujan's mathematics extensively is W. Backster, the well-known physicist from Australia. He used the famous Rogers-Ramanujan identities i n what is called the hard hexagon model to describe the molecular structure of a thin film.

Many of Ramanujan's works are used but his asymptotic formulae have found the most important application; I first wrote this in 1974 from his notebook.

Then there is a particular formula of Ramanujan's involving the exponential function which has been used many times in statistics and probability. Ramanujan had a number of conjectures in regard to this formula and one is still unproven. He made this con jecture in a problem he submitted to the Indian Mathematical Society. The asymptotic formula is used, for instance, in the popular problem: What is the minimum number of people you can have in a room so that the probability that two share a common birthd ay is more than half? I think it is 21, 22 or 23. Anyway, this problem can be generalised to many other types of similar problems.

Have you looked at the lost notebook?

That is what I am working on now with Andrews. It contains about 630 results. About 60 per cent of these are of interest to Andrews. He has proved most of these results. The other 40 per cent are of great interest to me as most of them were a continuatio n of what Ramanujan considered in his other notebooks. So, I began working on them.

What are your experiences of working on Ramanujan's notebooks? Do you think Ramanujan was a freak or a genius or he had the necessary motivation to write the notebooks?

I think one has to be really motivated to do the kind of mathematics he was doing, through either teachers or books. We understand from Ramanujan's biographers that he was motivated in particular by two books: S. L. Loney's Plane Trigonometry and Carr's Synopsis of Elementary Results in Pure Mathematics (which was a compilation of 5,000 theorems with a few proofs) at the age of 12. How much his teachers motivated him, we really do not know as nothing about it has been recorded. Reading these book s and going through the problems must have aroused the curiosity that he had and inspired him.

He is particularly amazing because he took off from the little bit he knew and extended it so much in so many directions, leading to so many new and beautiful results.

Did you find any results difficult to decipher in any of Ramanujan's notbooks?

Oh yes. I get stuck all the time. At times I have no idea where these formulae are coming from. Earlier, Ron Evans, whom I have already mentioned as having worked on Chapter 14, helped me out a number of times. There are times I would think of a formula over for about six months or even a year, not getting anywhere. Even now there are times when we wonder how Ramanujan was ever led to the formulae. There has to be some chain of reasoning to lead him to think that there might be a theorem there. But ofte n this is missing. To begin with, the formulae look strange but over time we understand where they fit in and how important they are than they were previously thought to be.

Did you find any serious errors in Ramanujan's notebooks?

There are a number of misprints. I did not count the number of serious mistakes but it is an extremely small number - maybe five or ten out of over 3,000 results. Considering that Ramanujan did not have any rigorous training, it is really amazing that he made so few mistakes.

Are the methods of mathematics teaching today motivating enough to produce geniuses like Ramanujan?

Some like G. E. Andrews think that much of the reforms have come about because students do not study as much. This, along with the advent of computers, has changed things. A lot of mathematics which can be done by computations, manipulations and by doing exercises in high school are now being done using calculators and computers. And the computer, I do not think, gives any motivation.

The books on calculus reform (that is now introduced in the U.S.) include sections on using a computer. To calculate the limit of a sequence given by a formula, the book says press these numbers, x, y and z... Then there appears a string of numbers that get smaller and smaller and then you can see that is tends to zero. But that does not lead to any understanding as to why they are tending to zero. So, this reasoning, motivation and understanding of why the sequence tends to zero is not being taught. I think that is wrong.

There seem to be two schools of thought: one which thinks that the development of concepts and ideas is important and the other, like that in India, which thinks that development of skills is important in teaching mathematics. Which do you think is m ore important?

I think you cannot have one without the other. Both must be taught. The tendency in the U.S. is to move away from skills and rely on computers. I do not think this is correct because if you have the skills and understanding, then you can see if you have made an error in punching in the computers. Andrews and I have the experience of students putting down results that are totally ridiculous because they have not understood what is going on. They do not even realise that they made mistakes while punching in the computers. So, developing skills is absolutely necessary. But on the other hand if you just go on with the skills and have no understanding of why you are doing this, you lose the motivation and it becomes just a mechanical exercise.

However, even now there is a possibility that geniuses like Ramanujan will emerge. It is important that once you identify such children, books and material should be found for them specially. The greatest thing about number theory in which Ramanujan work ed is that you can give it to people of all ages to stimulate them. Number theory has problems that are challenging, that are not too easy, but yet they are durable and motivating. A foremost mathematician (Atle Selberg) and a great physicist (Freeman Dy son) of this century have said that they were motivated by Ramanujan's number theory when they were in their early teens.

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