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George E. Andrews (Transcript of Video Recording on January 6, 2002. Interviewer: Dr. K. Srinivasa Rao. Place: Lucknow University, Venue for the Indian Science Congress 2002.) ******** KSR: Professor George Andrews, you are renowned as the discoverer of the ‘Lost Notebook of Srinivasa Ramanujan. I would like you to tell us how you came Across this Notebook in the Spring of 1976. GEA: Well, there is a great deal of just luck involved. In the year 1975 1976, I was on leave from Penn State, at the university of Wisconsin. In the spring of 1976, there was a conference in France that I was invited to attend and to speak up. At that time, the way airline fares were drawn up, if you stay in Europe for 3 weeks or more then the cost of the airline ticket was almost inconsequential. Anyway, for less than three weeks, the airline tickets were very expensive. So, I tried to figure out an academically sound program to mix in with this conference, in order to last 3 weeks and the university of Wisconsin graciously granted me the three weeks to be in Europe. One of the things that I wanted to do, that I listed up, that I have known about were papers from the estate of late G.N. Watson that were at the Trinity College Library. So, after the conference in Strasburg and speaking again in Paris, I then went to Cambridge just to look at these papers and to study them. To my surprise, these materials from the estate of Watson also included things from Ramanujan, the famous last letter he wrote to Hardy that had previously been published was in this box, as well as 137 pages of the unmistakable Ramanujan hand writing. Some on both sides of the paper. So, about 100 sheets and it was clear when looking at this that Ramanujan had written it in the last year of his life after he returned to India when he was very ill and dying. I say that because I had the good fortune to write my Ph.D thesis on mock Theta Functions and I found in this notebook all of the formulas on mock Theta Functions that Ramanujan had sent to Hardy even though he wrote no explanatory proofs, but none the less the formulas where there. That meant, once you saw that, you knew it was the material he had written in the last year of his life and consequently even though he was physically decaying, he was at the height of his mental powers and so, I knew that I had put my hands on the documents of immense mathematical interest. So, it was a grand event. KSR: There is one statement which has been made that it was neither ‘Lost’ nor was it a Notebook and that you had chosen these right words to make it very famous and another argument which was known at that time was that apparently the classification of all the papers of Ramanujan was made by Pro-

fessor Robert Rankin and Whittaker’s son and that they had definitely seen these documents. That was in the late Sixties. But you were the first person who saw that it was a Treasure Chest of formulae which should be studied in great detail. GEA: You see the advantage that I had which they did not have is that I had studied the mock Theta Functions and I was aware of them. But, if you did not know that, then you would assume a natural assumption might be that these were scratch notes related to the Notebooks that he had written prior to going to England. Unless you had that particular piece of information you would not date when this notebook was written and that was that it would have been none the less interesting but not nearly as interesting as if it had been more or less things that he had written in the construction of the note books prior to his going to England. So, yes both Rankin and Whittaker saw this and recognized it as something for which they advised Watson’s widow to contribute it to the Trinity College but they did not recognize its significance. So, when I called and saw that it was, of course, in the hands of very trusted English Men for 55 years previously. I say ‘Lost’ his intellectual document because no one knew what it was and when it was written and consequently did not know the importance of it. KSR: Now that you have said that it was mainly a fortuitous circumstance that you had studied the mock Theta Functions and you have been giving lectures on this ‘Lost’ Notebook for the past several years and your four articles that you have written – [SEVEN. I thought you had written four] – the seven articles you have written are pioneering articles and you are also responsible for releasing a copy of this in the centenary year 1987, you were gracious enough to release this volume to the public at Madras, on the occasion of Ramanujan’s Birth Centenary, and still I understand that you are editing this Notebook with Professor Bruce Berndt, obviously the last word has not been said on the contents of this Notebook or the valuation of this Notebook. The discoveries which are being made out of the entries seem to be continuing in this century. What would you like to say about the contents themselves ? GEA: I hold Ramanujan in great awe. The way he thought about mathematics is still in my mind mostly a mystery. I know that ever so often I think: may be, I penetrated to see the overarching view that he had. But I am always a little worried when I think that way. One experience in my life: In 1987, in one of the conferences on Ramanujan, I said that I thought I could explain how he thought about a list of identities. After the talk, the late K.G. Ramanathan came up to me and said well you know that actually Atle Selberg in his work looked at these identities and I think this is the way Ramanujan thought about it”. It may well have been so! Even when I thought I had figured out something, I 2

came to doubt about it. I tried to stress, in this Symposium today, the idea that it is a useful exercise to try to think of an overarching view of some of his achievements. While we will never have confirmation of how he thought about things, there seem to be general principles involved in the number of things he discovered. There may be alternative principles and in these likely most cases or in many cases we may never grasp how he thought about things. It is just a process of trying to see things in a more general setting where he has given certain explicit examples for something that has been immensely fruitful for me in my study of his work. Efforts to understand these things had led to a number of discoveries that I have made where I am uncertain whether he knew these things were right or not but none the less taking this point of view led to these things and I think that especially with the work that Bruce Berndt has done in his editing of the famous 3 notebooks and you know that we are editing this ‘Lost’ notebook. This is certainly a first stage of things. We will have proofs of almost all, if not all, of the formulas but the real understanding of them is some distance off. Even today, in this symposium, this discussion of mock Theta Functions it becomes clear from the things that people have said, that no real overarching understanding exists at the moment and yet there is so much structure that it is obvious at the ground level that this makes it a topic that is of further significant interest. KSR: It has been stated that Ramanujan was probably working towards a big theory, that he was only discovering or revealing only parts of his discoveries in his Notebooks. Is there a kind of feeling that probably he was dealing with a bigger structure of things ? GEA: Well, let me give you one hint. There are a couple of cases in his ‘Lost’ Notebook in which he sets up a basic frame work for dealing with the third order of Mock Theta Functions and then in the following page he sets up a perfectly parallel framework to find out a sort of overarching functions from which he reduces several formula so then he goes to third order then goes to fifth order then down at the bottom of this page there is exactly the same overarching function for the seventh order and then he puts in a equal sign and then you turn the next page there is nothing there related to the things that been went before and you non the less feel somehow that he either got stuck or there are pages missing. At least the hint there is he I believe he believed that it was the beginning of something but I dont know. He is obviously excited from the words that he wrote to Hardy it is clear that he thought that this was a topic of significance but clearly this was very late in the day for him. He was months away from dying and if he had gone more into this, we have not seen it. It has disappeared. KSR: One thing that I would like to ask is that for creativity there is no peer to 3

Ramanujan. Atle Selberg, in his Tata Institute lecture in 1987 made a statement that conjectures are the things which are a source of lot of creativity for other mathematicians. I would like you to comment on the creativity of Ramanujan, vis-a-vis the giants with whom he has been compared, like Euler, Gauss, Jacobi and others. GEA: Well, I would first comment on the relationship between conjecture and proof. Obviously to make sweeping statements that are probably unwise but in terms of what Ramanujan did in terms of Notebook and what we see there is a list of formulas without proof and so is natural to look at them as conjectures although it is not necessarily that these were the conjectures that Ramanujan made. He had a proof or an outline of a proof written on a slate and when he choose to write the formula down or he may not have! There are a certain formulas in this lost note book where he will put a question mark after a formula and at least one of the question marks was wrong. So that suggests that at least in those cases he did not have the proof but was thinking it was at least worth recording. It is difficult for me to evaluate Ramanujans creativity by something I dont know whether I want to say usually where I think of creativity in terms of things that are not sort of the undiscovered truths of mathematics. I see Ramanujan as some one who has this immense genius who was capable of grasping proofs in ways that seem to be beyond the average mode in thinking about them and certainly, Littlewood virtually when first informed of Ramanujans work in his first letter to Hardy he said: “I think he is at least a Jacobi”. It is always I think a dangerous game to rank the great mathematicians. We heard several times today that Hardy had stated that on a ranking for zero to hundred, he gave Ramanujan a hundred and Hilbert had eighty and Littlewood at thirty and Hardy himself at twenty five and I think that it more or less betrays extreme confidence on Hardys part, which was perhaps a little unjustified. We do recognize the great creativity of Ramanujan had. Hilbert was absolutely a towering figure in Mathematics. To say that he was 80I don’t think I want to hassle. But Ramanujan, the uniqueness of Ramanujan is something that is an inspiration to us all because we are so very proud of him. After I finished the conference in Stasburgh and wanted to examine Watsons Papers, I went to the Trinity College Library where I knew they were, and here was the marvellous find of the 137-page document from Ramanujan, with Ramanujans handwriting on Mock Theta Functions. So, it was obvious to me, that it was a document that was to sit permanently in the Trinity College Library and I needed desperately a copy of it. The Trinity College Library is a very austere and intimidating institution. It is not like a modern American Media Centre. This is a place where a bust of Newton stares down at you and bunch of people stare down at you that you do no damage. So, I went up to the desk and nervously asked if I could have a Xerox copy of. It was like asking a 4

Xerox of the writings of Saint Paul. But, the woman behind the desk was very friendly and said: “Well, of course, I will be glad to Xerox it for you and we will send it right off today and it would be ready only in two weeks”! I knew of British inefficiency but this seemed unusual that it would take two weeks to Xerox 130 pages. None the less, whatever conditions that they wanted to lay down was fine. So, I said fine. When you finish it, I will like you to Air Mail to me at the University of Wisconsin, for which she said, “Oh! My Goodness, that would be very expensive.” As I had said many times before, I was prepared to take a second mortgage of my house to get a copy of this. For this she said“nine pounds” and I thought nine pounds was nothing. So, fine, I will take it. So everything was arranged and I got a very perfect address and I only had two more days in England and so I wanted to do some other things so I came back the next day and there sitting on my desk was a large mailing envelop and my name was on it carefully written out and every thing. So, I knew that the British Xerox was better than she had suggested and there was my ‘Lost’ Notebook, sitting on her desk. But now she was not there. Someone else was there who was confident that this thing was to be mailed and put in the post. With great effort I convinced him that actually I was the person this was addressed to and it would be alright if he just gave it to me and I left with the ‘Lost’ Notebook. So this was a discovery that I knew that I would be the first in the world and the second most excited person would be my colleague back at the University of Wisconcin, Prof. Richard Askey. So, when I returned I talked to him on the phone he said is there that anything of interest for me from the trip and I said, yes, I will bring to him tomorrow a 130-page document written in the last year of Ramanujans life in his own handwriting and “you can look at it at 25 cents a glance!

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