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Ramanujan's mentor

Today is Ramanujan's 115th birth anniversary. To mark the occasion, KRISHNASWAMI ALLADI describes the life and contributions of the British mathematician Hardy, and discusses his collaboration with Ramanujan.

Ramanujan... Ranked 100 on a scale of 1-100 by Hardy.

G. H. HARDY, a towering figure in analysis and number theory, had written several important research papers and influential textbooks on these subjects. When Ramanujan wanted to get the opinion of British mathematicians to evaluate his discoveries which lay at the interface between analysis and number theory, it was only natural that he close to write to Hardy.

Actually Ramanujan communicated his remarkable findings to several British mathematicians, but it was only Hardy who responded. Realising that Ramanujan was a genius of the first magnitude who would profit immensely by contact with professional research mathematicians, Hardy invited Ramanujan to Cambridge University, England. The rest is history. The collaboration between Hardy and Ramanujan, the influence they had on each other, and the impact their work had over mathematicians of their generation and those succeeding them, was immense.

Early life and education: G. H. Hardy was born on February 7, 1877 in Cranleigh, in Surrey, England as the elder of two children. His father was a schoolteacher at Cranleigh School and his mother a Senior Mistress at a training college. Although both parents were mathematically minded, they did not attend university owing to financial reasons. Both Hardy and his sister were encouraged to read good literature and discover things for themselves. Hardy showed precocity with arithmetic and passed examinations with distinction in mathematics and Latin at Cranleigh School. He then secured a scholarship at Winchester College, but was not satisfied with the education there. Although he intended initially to go to Oxford University from Winchester, he changed his mind and joined Trinity College at Cambridge University in 1896.

Cambridge education: At Cambridge Hardy started to train for the famous Tripos exams. These exams were undoubtedly among of the most severe mathematical tests. Although Hardy was successful in solving the Tripos problems, he hated these exams and what they stood for. In fact Hardy was much irritated with his Tripos coach Dr. Webb. Fortunately, the Director of Studies put Hardy under the care of the applied mathematician Augustus Love who asked Hardy to read Camille Jordan's wonderful treatise Cours d' Analyse. Hardy says "My eyes were opened by Professor Love who taught me for a few terms and gave me the first serious conception of analysis. I shall never forget the astonishment with which I read the remarkable work (of Jordan)... "Just as Hardy detested the formality of the Tripos, Ramanujan was uninspired by the dull college curriculum. Whereas Hardy was inspired by Jordan's rigorous treatment of analysis, which influenced his own style of research, Ramanujan was influenced by Carr's synopsis, which according to his Indian biographers was a treasure house of delightful formulas that awakened his genius. Unlike Ramanujan, who dropped out of college, Hardy held on to be successful at the Tripos exams in 1900 and was in fact placed in the first division of First Class. Hardy was elected to a prize fellowship at Trinity College and in 1901 won the prestigious Smith's prize along with the physicist James Jeans.

Influential papers and books: It was in 1900 that Hardy began to publish his mathematical discoveries. His first paper on definite integrals appeared in the Messenger of mathematics. Hardy subsequently published more than 60 papers in the theory of integration over the next three decades. Between 1905 and 1915 Hardy wrote four books in the Cambridge Mathematical Tracts series on The Integration of Functions of a Single Variable (1905), Orders of Infinity (1910), The General Theory of Dirichlet Series (with Marcel Riesz in 1915), and what was to become his most well known book A Course in Pure Mathematics (1908). During the post Newtonian era, although there were many no-table mathematical contributions from England, British mathematics had taken a backseat compared to the phenomenal contributions from mainland Europe. Hardy and a few other British mathematicians, notably, J. E. Littlewood, Hardy's collaborator in Cambridge, were leading the British revival in mathematics. Hardy was an especially polished and prolific writer and had the greatest effect in leading this resurgence of British mathematics. His books and papers had a powerful influence not only on mathematicians in England but overseas as well. Indeed, Hardy's book Orders of Infinity attracted Ramanujan who was intrigued by the problem of determining the number of primes below a given magnitude, and it was this that prompted Ramanujan to write to Hardy.

Ramanujan's letters: The two letters Ramanujan wrote to Hardy in 1913 are considered to be among the greatest in mathematical history. The letters not only contained a fantastic collection of spectacular formulae, but also profoundly influenced the mathematical careers of both Hardy and Ramanujan. Hardy's initial reaction on seeing the letters was that Ramanujan was a fraud because many of the formulas were known, some were incorrect, and there were no hints of proofs. But then there were several astonishingly beautiful formulas that were correct and very deep. Only a mathematician of the highest class could write them down. So, on second thought, Hardy concluded that it was more probable that Ramanujan was a genius and unlikely that he was a fraud because no one but a true genius could have the imagination to invent such formulae. The great philosopher Bertrand Russell says that one evening in Trinity College he found the usually placid Hardy in a wild state of excitement talking about a new Euler or Jacobi from India! Hardy was convinced that Ramanujan was wasting his time in India rediscovering past work, and would profit immensely by coming into contact with professional mathematicians.He thus invited Ramanujan to Cambridge University to work with him. Although Ramanujan's mother initially resisted this, she eventually realised that she should not stand in the way of her son's progress, and so gave Ramanujan permission to go to England.

The Hardy-Ramanujan interaction: Ramanujan spent only a few years in England (1914-19), and although he was ill quite often in this brief period, he wrote several fundamental papers, some jointly with Hardy. These papers have had a lasting influence on various branches of mathematics and led to surprising connections between fields considered quite distinct. When Ramanujan was well, he met Hardy almost every day and was showing his mentor a dozen new formulas at every meeting. It was difficult for Hardy to keep up with Ramanujan's progress and leaps of imagination. Indeed Hardy admitted that a simple supervisor was inadequate for so fertile a pupil! Yet, the raw genius of Ramanujan combined with the scholarship of Hardy produced magnificent results. Formula for partitions: In one of the letters to Hardy in 1913, Ramanujan gave a formula for the coefficients of a certain series expansion of an infinite product which suggested that there ought to be a similar exact formula for p(n), the number of partitions of a positive integer n. Hardy felt that this claim of Ramanujan concerning an exact formula for p(n) in terms of continuous functions was too good to be true but was convinced that it would be possible to construct an asymptotic series expansion. Asymptotic formulas are like approximate formulas. By an ingenious and intricate involving the singularities of the generating function of p(n), Hardy and Ramanujan obtained an asymptotic formula which when calculated up to a certain number of terms yielded a value that differed from p(n) by no more than the fourth root of n. Since p(n) is an integer, it is clear that its value is the nearest integer to what is given by the series. The proof of this formula required sophisticated tools from complex variable theory and here Hardy's mastery over analytic methods was crucial. The series Hardy and Ramanujan obtained was genuinely an asymptotic series in the sense that when summed to infinity it diverges. Subsequently, Hans Rademacher noticed that by making a very mild but important change, namely, by replacing the exponential functions by hyperbolic functions, the Hardy-Ramanujan asymptotic series could in fact be converted to an infinite series that converges to p(n). Actually in the 1913 letter to Hardy, Ramanujan used hyperbolic functions to claim an exact formula for a related problem. So Ramanujan was indeed correct in surmising that a similar exact formula would exist for p(n).. Professor Atle Selberg of the Institute for Advanced Study in Princeton, one of the greatest living mathematicians, said during his talk at the Ramanujan Centennial in Madras in 1987, that the exact formula of Rademacher was indeed more natural than the Hardy-Ramanujan asymptotic formula. Selberg discovered this formula on his own but did not publish this once he found out that Rademacher had obtained it earlier. According to Selberg, even though Ramanujan correctly claimed the existence of an exact formula for p(n), out of respect for his mentor Hardy, he settled for less, namely the asymptotic formula. In any case, the Hardy-Ramanujan asymptotic formula gave rise to a very powerful analytic method to evaluate the coefficients of series arising in a wide class of problems in additive number theory. This circle method, originally due to Hardy-Ramanujan, was subsequently developed by Hardy and Littlewood and others, and is one of the most widely applicable methods today.

Round numbers: Hardy and Ramanujan were interested in a mathematical explanation of the phenomenon round numbers are rare, where loosely speaking, a number is round if it is composed of a large number of relatively small prime factors. So, they went about studying the behaviour of w(n), of the number of prime factors of a positive integer n. It is a curious fact of history that although prime numbers have been studied since Greek antiquity, it was only in the beginning of the 20th Century with the work of Hardy and Ramanujan that the number of prime factors of n was studied systematically. Hardy and Ramanujan showed that on average w(n) was about loglog n in size, and that this average also indicated the size of w(n) almost always. This was somewhat surprising since for most number theoretic functions, the average is not a true indication of the size, because the average is often influenced by large values which occur infrequently. In the case of w(n), the large values had no influence on the average because they occurred with very low frequency. Thus they explained that round numbers, which are numbers for which w(n) is large, are rare. The true significance of the Hardy-Ramanujan observation on round numbers was not realised until a few decades later, when Paul Turan, Paul Erdos, and Mark Kac showed the probabilistic underpinnings of this result. Indeed, with the Hardy-Ramanujan work on round numbers as the foundation point, and the beautiful superstructure built on it by Turan, Erdos and Kac, the subject of Probabilitic Number Theory was born, and is a very active field on research today.

Honours for Ramanujan: Hardy was convinced that since Ramanujan's work was phenomenal, he deserved to be honoured by the Royal Society. But time was running out. Ramanujan was getting ill more often and was frequently in and out of nursing homes. At one point Ramanujan was so depressed he attempted suicide. There is a story that when the police wanted to question Ramanujan on the attempted suicide, Hardy intervened and convinced the police not to go through with this by telling them that Ramanujan was a Fellow of the Royal Society (FRS) — which he was not at that time! Concerned about the continued illness, Hardy arranged for Ramanujan to return to India where he felt the care would be better with the help of kith and kin. But Hardy wanted Ramanujan to get the honours before his return to India for two reasons. First, he felt that the honours would boost Ramanujan's spirit and could have a positive effect on his health. Second, Hardy feared that Ramanujan may not live too long. Hardy worked very hard to successfully convince his colleagues in the Royal Society, and so Ramanujan was elected FRS in 1918. Ramanujan was also made Fellow College that same year. Although Hardy was saddened to see Ramanujan leave for India in 1919 as a sick man, he said that everyone should be proud that Ramanujan was returning to his homeland in glory and with a reputation that transcended all human jealousies

Hardy was a towering figure in the world of mathematics.

Move to Oxford: The departure of Ramanujan in 1919 left a gap in Hardy's life. He was also dissatisfied that his presence was not sufficiently appreciated in Cambridge. So he accepted the Savillian Chair of Geometry at Oxford University in December that year. Hardy's close friend and collaborator J. E. Littlewood succeeded his as Cayley Lecturer in Cambridge. As professor at Oxford, Hardy had much freedom to lecture on topics to suit his tastes and schedule. Since the Savillian Professorship was for geometry, he lectured on that subject in Oxford, and gradually added number theory and the theory of functions to his lectures. Hardy humorously remarked "I do not claim to know any geometry, but I do claim to understand quite clearly what geometry is." Hardy's eleven years at Oxford were the happiest of his life. He was taken much more seriously at Oxford than at Cambridge. Colleagues used to look forward to his presence at the afternoon teas and other occasions, where he was the centre of attention. While at Oxford, his collaboration with Littlewood continued unabated.

Return to Cambridge: In 1931 Hardy accepted the Sadlerian Chair at Cambridge, which for many years was occupied by the great mathematician Arthur Cayley, and also briefly by his former professor Augustus Love. At Cambridge Hardy was allowed to stay in the rooms at Trinity College, whereas there, were no such similar facilities at Oxford. By the time Hardy returned to Cambrdige, Littlewood was appointed to the Rouse Ball Professorship there. Although Hardy had created a school in analysis at Oxford that was second to none, Cambridge was the world's centre for mathematics, and so it was only appropriate that he returned to spend the rest of his life.

Honours for Hardy: The 1920's were years of honours both Hardy and Littlewood. Hardy was awarded the Royal Medal of the Royal Society in 1920. He served as Secretary of the London Mathematical Society from 1917 to 1926 and never missed a single meeting of the Society. He was President of the Society during 1926-28, and received the Society's highest honour, the De Morgan Medal in 1929. In 1926 Hardy founded the Journal of the London Mathematical Society, and subsequently a new series of the Oxford Quarterly Journal of Mathematics. Ironically, Hardy died on 1 December 1947, the very day he was to be presented the Copley Medal of The Royal Society.

Eccentricities, habits, and beliefs: Hardy was a bachelor all his life and was wedded to mathematics. He was an avid tennis player and loved to watch cricket. He was a sworn atheist. He would never wear a watch or use a fountain pen, and hated to have his picture taken. On one occasion he gave the following list of New Year resolutions: 1) Prove the Riemann Hypothesis, 2) make 211 not out in the fourth innings of the last test match at the Oval, 3) find an argument of the non-existence of God which would convince the general public, 4) be the first man on top of Mt. Everest, 5) be proclaimed the first president of USSR, Great Britain, and Germany, and 6) murder Mussolini.

Estimation of Ramanujan: Hardy had the highest admiration for Ramanujan. He said that his contact with Ramanujan was the one romantic incident of his life. He was also felt fortunate that he was the only person who "collaborated with Littlewood and Ramanujan on somewhat equal terms." When invited to speak at Harvard University for the Tercentenary Celebrations, he chose to speak about Ramanujan's work. Hardy's 12 lectures on Ramanujan based on those given at Harvard, are models of lucidity in exposition. On one occasion in ranking mathematicians on the basis of pure talent, he gave Ramanujan the highest score of 100. On this scale, he gave the great German mathematician Hilbert a score of 80. Hardy gave himself a score of only 25, but said his colleague Littlewood merited 30 since he was the more talented of the two.

During the Ramanujan Centennial in Madras in 1987, Atle Selberg speculated that the great German mathematician Hecke would have been a better mentor for Ramanujan than Hardy. Selberg's observation was based on the fact that many of Ramanujan's most significant discoveries lay in the area of modular forms, where Heke was an expert, and an area, which was not Hardy's cup of tea.

But there are other factors that are crucial, such as a common language of communication, willingness to spend time with the pupil, and above all, mutual respect. Hardy did everything he could to encourage Ramanujan. The above the rankings summarise best Hardy's modesty, and his respect for Ramanujan. Both Hardy and Ramanujan blossomed as a consequence of their interaction.

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