On a height he stood that looked towards greater heights.
Our early approaches to the Infinite
Are sunrise splendours on a marvellous verge
While lingers yet unseen the glorious sun.
What now we see is a shadow of what must come.
Sri Aurobindo, Savitri, 1.4
The story of Srinivasa Ramanujan is a 20th century “rags to mathematical riches” story. In his short life, Ramanujan had a wealth of ideas that have transformed and reshaped 20th century mathematics. These ideas continue to shape mathematics of the 21st century. This article seeks to give a panoramic view of his essential contributions.
Born on December 22, 1887 in the town of Erode in Tamil Nadu, Ramanujan was largely self-taught and emerged from extreme poverty to become one of the most influential mathematicians of the 20th century. How did this transformation come about? Though it is difficult to pinpoint any precise causes for this transformation, one can delineate several significant events in his life that enabled this to come about.
Ramanujan cultivated his love for mathematics singlehandedly and in total isolation. As a child, he was quiet and often kept to himself. Those that knew him were impressed by his shining large eyes, which were his most prominent features. He had a prodigious memory, and at school he would entertain his friends by reciting the various declensions of Sanskrit roots, and by repeating the value of the constant ‘pi' to any number of decimal places. This was a foreshadow of what was to come, since later in life he would write a monumental paper that would connect the computations of the digits of ‘pi' to modular forms, a theory developed largely in the 20th century. It is a theory which is definitely at the forefront of modern mathematics today and we will expand on this theme later in this article.
At the age of 12, he borrowed from a friend a copy of Loney's book on Plane Trigonometry, published by Cambridge University Press in 1894. This book goes far beyond high school trigonometry and also deals with the rudiments of calculus. But the book that changed his life was Carr's book titled, A Synopsis of Elementary Results in Pure and Applied Mathematics . This book is a compilation of 6,165 theorems, systematically arranged but with practically no proofs. It is not a remarkable book, and Ramanujan's use of it to propel himself to the centre stage of 20th century mathematics, has made the book remarkable. It was largely used by students of Carr who were preparing for the entrance examination in mathematics at Cambridge University. Ramanujan used the book to master all of 18th and 19th century mathematics. He set about to demonstrate each of the assertions of the book, using only his slate to do the calculations. He would jot down the formula to be proved, and then erase it with his elbow, and then continue to jot down some more formulas. In this way, he worked through the entire book. People used to speak of his “bruised elbow.” Sadly, he took Carr's book as a model for mathematical writing and left behind his famous notebooks containing many formulas but practically no proofs. Many mathematicians have made it an industry to prove these formulas that Ramanujan had scribbled into his notebooks since he left no hint as to how he got them.
In 1903, Ramanujan entered the Government College in Kumbakonam. Unfortunately, he failed in the examination since he neglected his non-mathematical subjects.
Four years later, he entered another college in Chennai, and the same thing happened. Finally, in 1912, he secured a job as a clerk in the Madras Port Trust Office. Here, his duties were light and so he could devote a lot of time to his mathematical discoveries — which he recorded in his now celebrated notebooks. As luck would have it, the manager of the office, S.N. Aiyar, was also a mathematician who took kindly to Ramanujan and encouraged him in his mathematics. It was he who suggested to Ramanujan that he write to G.H. Hardy, a famous mathematician at Trinity College, Cambridge University.
In his famous 1913 letter to Hardy, Ramanujan attached 120 theorems as a representative sample of his work. Some of these formulas Hardy had already seen in the course of his own research work. But many of the other formulas, he had not. It took over two hours for him to analyse the letter in order to determine if it was written by a crank or a genius. He consulted with his eminent colleague J.E. Littlewood, also of Trinity College, and together they sat down for three more hours. Finally they concluded that it was the work of a genius. Hardy wrote: “They must be true, because if they were not true, no one would have had the imagination to invent them.'' With this certificate of approval, Ramanujan was invited to come to Trinity College to work with Hardy.
Ramanujan sailed to England in March 1914, just a few months before the outbreak of the First World War. From 1914 to 1917, Hardy and Ramanujan collaborated on more than half a dozen research papers. At the same time, Ramanujan published more than 30 research papers in three years. The most notable of these collaborations involved the partition function. This function counts the number of ways a natural number can be decomposed into smaller parts. Hardy and Ramanujan developed a new method, now called the circle method, to derive an asymptotic formula for this function. If one analyses Ramanujan's first letter to Hardy, we already find a hint of the method in his work done in India while at the Port Trust Office. This method is now one of the central tools of analytic number theory and is largely responsible for major advances in the 20th century of notoriously difficult problems such as Goldbach's conjecture, Waring's conjecture and other additive questions. The circle method and its refinements constitute a very large area of current research and will probably continue to be so in the 21st century.
Another fundamental paper of Hardy and Ramanujan concerns what is now called the “normal order method.'' This method analyses the behaviour of additive arithmetical functions. In their paper, Hardy and Ramanujan showed that a random natural number usually has about log log n prime factors. Their paper led to the creation of an entirely new field of mathematics called probabilistic number theory. In the 20th century, it was largely developed by P. Erdos, M. Kac and J. Kubilius.
But the paper that really changed the course of 20th century mathematics was the one written by Ramanujan in 1916, modestly titled “On certain arithmetical functions.'' In this paper, Ramanujan investigated the properties of Fourier coefficients of modular forms. At that time the theory of modular forms was not even developed. However, Ramanujan enunciated three fundamental conjectures that served as a guiding force for the development of the theory.
Indeed, the first two of his conjectures led to the development of what is now called Hecke theory, formulated by E. Hecke in 1936, twenty years after Ramanujan's paper. Many would have heard of Fermat's last theorem and how this was solved in 1994 by A. Wiles. But few will know that Wiles used Hecke's theory in an essential way in his solution of the problem.
However, it was the last of the three of Ramanujan's conjectures that created a sensation in 20th century mathematics. This conjecture, later called Ramanujan's conjecture, came to play a pivotal role in the towering edifice known as the Langlands program, a far-reaching program articulated by R.P. Langlands in the 1970s. This program connected two seemingly different fields of mathematics, namely representation theory and number theory. But the proof of Ramanujan's third conjecture came about through another route connecting algebraic geometry to number theory in the framework of general conjectures of A. Weil concerning the number of solutions of equations over finite fields. The Weil conjectures were settled by P. Deligne in 1974 and he was awarded the Fields Medal (the mathematical equivalent of the Nobel Prize) for this work. Ramanujan's third conjecture turned out to be a special case of the Weil conjecture. Ramanujan's conjecture is now seen as a spectral line of a larger spectrum of conjectures, now called the generalised Ramanujan conjecture.
Last letter to Hardy
If Ramanujan's 1916 paper created a sensation by heralding the development of the theory of modular forms, his last letter to Hardy, written literally on his deathbed in 1920, outlining a new theory of “mock theta functions,” is now creating a greater sensation in the development of 21st century mathematics. Indeed, Ramanujan's theory of mock theta functions was largely ignored for much of the 20th century and was discussed in sporadic papers. Part of the difficulty was with Ramanujan's vague definition of a mock theta function. In fact, he never defined them. Rather, he listed 17 protypical examples of these new functions and formulated general conjectures concerning them. Many mathematicians tried to prove these conjectures without a proper theory in place. To a large extent, they succeeded in proving most of Ramanujan's conjectures. However, the unifying conceptual framework was missing. This framework was discovered only recently in 2002 in the doctoral thesis of S. Zwegers, written under the direction of D. Zagier. This thesis laid the groundwork for a new theory of mock modular forms.
We now understand Ramanujan's theory of mock theta functions as a special case of a larger theory of mock modular forms. These objects are generalisations of modular forms and thus include the classical theory of Hecke as a special case. Already, the richer theory of mock modular forms is bearing new mathematical fruit, as is evidenced by some recent breakthrough works of J. Bruinier, J. Funke, K. Bringman, and K. Ono. For instance, Bruinier and Ono recently derived an algebraic formula for the partition function using the theory of mock modular forms. M. Dewar and R. Murty noticed that this Bruinier-Ono formula can be used to derive the Hardy-Ramanujan formula for the partition function and thereby avoid the complicated circle method. These new viewpoints are definitely the tip of the iceberg, concealing a larger mass of mathematical truth.
In 1987, the famous physicist, Freeman Dyson, predicted: “The mock theta functions give us tantalising hints of a grand synthesis still to be discovered. It should be possible to build them into a coherent group-theoretical structure, analogous to the structure of modular forms which Hecke built around the old theta functions of Jacobi. This remains the challenge for the future.”
Indeed, Dyson's prediction is right on target. The recent advances in the theory are just a foreshadow of greater things to come. Once the theory of mock modular forms is in place, it is only a question of time to marry the theory to the larger program of Langlands. This may be delicate, and one should not go too fast lest we miss the scenic beauty along the route. Nevertheless, it is the direction of the future. Thus, Ramanujan's work has had a fundamental role in the development of 20th century mathematics and his final writings are serving as an inspiration for the mathematics of this century.
We do not know how Ramanujan discovered his theorems. On this point, Hardy wrote: “It was his insight into algebraic formulae, transformations of infinite series and so forth, that was most amazing. On this side most certainly I have never met his equal, and I can compare him only with Euler or Jacobi. He worked far more than the majority of modern mathematicians, by induction from numerical examples; all his congruence properties of partitions, for example, were discovered in this way. But with his memory, his patience and his power of calculation, he combined a power of generalisation, a feeling for form, a capacity for rapid modification of his hypothesis, that were often really startling, and made him, in his own peculiar field, without a rival in his day.”
But beyond the mathematical legacy, Ramanujan left behind a cultural legacy. He appeared in the midst of the British colonial rule of India and now stands as an iconic symbol of an India that was rediscovering itself, an India that was rising up to take its place in the 20th century. This meant that science and education were to be revived and energised to meet the challenges of the new, independent India. Ramanujan's role in such a revival is best described in the words of Nobel laureate Subramanyam Chandrasekhar who, on the occasion of Ramanujan's birth centenary in 1987, wrote: “It must have been a day in April 1920, when I was not quite ten years old, when my mother told me of an item in the newspaper of the day that a famous Indian mathematician, Ramanujan by name, had died the preceding day; and she told me further that Ramanujan had gone to England some years earlier, had collaborated with some famous English mathematicians and that he had returned only very recently, and was well-known internationally for what he had achieved. Though I had no idea at that time of what kind of a mathematician Ramanujan was, or indeed what scientific achievement meant, I can still recall the gladness I felt at the assurance that one brought up under circumstances similar to my own, could have achieved what I could not grasp. I am sure that others were equally gladdened. I hope that it is not hard for you to imagine what the example of Ramanujan could have provided for young men and women of those times, beginning to look at the world with increasingly different perceptions. The fact that Ramanujan's early years were spent in a scientifically sterile atmosphere, that his life in India was not without hardships, that under circumstances that appeared to most Indians as nothing short of miraculous, he had gone to Cambridge, supported by eminent mathematicians, and had returned to India with every assurance that he would be considered, in time, as one of the most original mathematicians of the century — these facts were enough, more than enough, for aspiring young Indian students to break their bonds of intellectual confinement and perhaps soar the way that Ramanujan did.”
In these words of Chandrasekhar, we see the remarkable legacy left behind by Ramanujan. For the life of Chandrasekhar was equally full of hardships. Born in the same village surroundings as Ramanujan, he went to study at Cambridge and became a leading astrophysicist of the 20th century, finally being awarded the Nobel Prize in 1983. Indeed, he soared the way Ramanujan did.
But a scientist belongs to no nation. Many scientists from around the world have testified that they gained inspiration from the life story of Ramanujan. For Ramanujan embodies that marvellous miracle of the human mind to frame concepts and to use formulas and symbols as tools of thought to probe deeper into the mysteries of the universe, and the mysteries of one's own being. As long as the spirit of inquiry is alive, his legacy will pass from one generation to the next.
(M. Ram Murty is Professor and Head of the Department of Mathematics at Queen's University in Kingston, Ontario, Canada. V. Kumar Murty is Professor and Head of the Department of Mathematics at University of Toronto, Toronto, Ontario, Canada.)