My Search for Ramanujan

by Ken Ono

  That same year, 1912, Janaki finally came to Madras to live with the man who was formally her husband. He had a position now, and could support her, and she was thirteen, on the cusp of adolescence. Komalatammal came from Kumbakonam to live with the couple, further training Janaki in the occupation of housewife. She was a constant chaperone, and when she had to be absent, another woman would take her place in the household.

  Ramanujan had obtained his post at the Port Trust through a recommendation by Ramachandra Rao, his staunch patron in Nellore. His supervisors in Madras were the Englishman Sir Francis Spring and his subordinate, the Indian S. Narayana Iyer. His work, which consisted in reviewing accounts of the trust, allowed him time to pursue his mathematical research, and both his supervisors encouraged him in that direction. For the first time in his life, Ramanujan had satisfactory employment, was appreciated both for his performance at work and his mathematical endeavors, and could devote time to filling his curious notebooks with a treasure of mathematical identities. But both Sir Francis and Narayana Iyer understood that Ramanujan needed an audience for his work outside of India, since no Indian mathematician could fully understand and appreciate his work, and as an Indian, Ramanujan would unlikely receive recognition by the British establishment in India, which maintained a strict color bar. They decided to appeal to mathematicians in Britain on Ramanujan’s behalf.

  They began by writing to Micaiah John Muller Hill, Astor Professor of Mathematics at the University of London, who had served as chancellor of the university and was a Fellow of the Royal Society. Hill wrote back courteously, suggesting that Ramanujan be more careful with his notation and the presentation of his results, but he had clearly failed to appreciate Ramanujan’s genius, which was buried in his nonstandard notation and paucity of proofs. Disappointed, Ramanujan himself wrote to two British mathematicians, H.F. Baker, a Fellow of the Royal Society and former president of the London Mathematical Society, and E.W. Hobson, also a Fellow of the Royal Society and chair of pure mathematics at Cambridge University. Both returned Ramanujan’s submissions without comment.

  Ramanujan did not give up. On January 16, 1913, he wrote to G.H. Hardy, the Sadleirian Professor of Mathematics at Cambridge University and also a Fellow of the Royal Society. Unlike Ramanujan, who loved only mathematics, Hardy was interested in many subjects. He had almost decided to study history on entering university; he was interested in music and art; and throughout his life, he was fanatically devoted to the game of cricket

  Hardy viewed himself as a “pure mathematician” and boasted that his work had no application to the real world. (Little did he know that one of his main interests, the theory of prime numbers, would become the basis of the encryption algorithms that have become omnipresent in the digital age.) In his 1940 essay A Mathematician’s Apology, Hardy expounded his views about the importance of the purely intellectual pleasures of mathematics, stripped of any direct application to the real world. He compared proofs of theorems to poems, works of art, and musical compositions.

  Hardy was the leading number theorist in the world at that time, and his textbook on number theory, An Introduction to the Theory of Numbers, written with E.M. Wright, would become a classic in the field. Moreover, in 1911, Hardy had begun a long and strikingly fruitful mathematical collaboration with his colleague John E. Littlewood, a titan in the field of mathematical analysis. This was a pair eminently qualified to judge Ramanujan’s work, and in writing to Hardy, Ramanujan had aimed his dart well.

  G. H. Hardy

  Madras Port Trust Office

  Accounts Department

  16th January 1913

  Dear Sir,

  I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of only £20 per annum. I am now about 23 years of age. I have had no University education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at Mathematics. I have not trodden through the conventional regular course which is followed in a University course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as “startling.”

  There followed pages of formulas and assertions, all offered without a hint of proof. Ramanujan touched on divergent series, exotic integrals, formulas involving π, hypergeometric functions, prime numbers, divisor functions, sums of squares, and continued fractions. He considered the series of positive integers, namely , and he claimed that it could be interpreted as summing to −1/12 instead of infinity, hinting at his rediscovery of results of Bernoulli, Euler, and Riemann. He offered remarkable-looking formulas involving π and the trigonometric functions sine and cosine. Some of his formulas were incorrect, but the errors could be corrected once the idea was understood. Some formulas remained unproved for a long time, while others await a solution by future mathematicians. Some of Ramanujan’s expressions are so abbreviated or cryptic that mathematicians today continue to struggle over them.

  Ramanujan concluded his letter with a heartfelt request:Being poor, if you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the expressions that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you give me. Requesting to be excused for the trouble I give you.

  I remain, Dear Sir

  yours very truly

  S. Ramanujan

  © Springer International Publishing Switzerland 2016

  Ken Ono and Amir D. AczelMy Search for Ramanujan10.1007/978-3-319-25568-2_13

  13. These Formulas Defeated Me Completely

  Ken Ono1 and Amir D. Aczel2

  (1)Department of Mathematics and Computer Science, Emory University, Atlanta, GA, USA

  (2)Center for Philosophy & History of Science, Boston University, Boston, MA, USA

  And so it was that toward the end of January 1913, or perhaps in the first days of February, Hardy, over breakfast in his rooms at Trinity College, in Cambridge, received a curious letter from India containing nine pages of Ramanujan’s original mathematical work.

  A mathematician of Hardy’s rank and reputation often received letters from amateurs who thought they had discovered something new in mathematics and were seeking validation, perhaps publication, perhaps even renown. As an editor of several mathematical journals, I too am accustomed to receiving manuscripts from amateurs. Each year, I receive dozens of flawed proofs of famous problems, such as Fermat’s last theorem and the Riemann hypothesis, that have stumped generations of mathematicians. (Fermat’s last theorem, having resisted solution for almost four hundred years, was finally proved by Andrew Wiles in 1995 using almost the entire armamentarium of advanced twentieth-century mathematics, while the Riemann hypothesis continues to defy the world’s greatest mathematical minds.)

  Most of these invalid proof attempts fail because the amateur has made a false assumption that if true would render the problem trivial. It’s rather like getting trapped in a fool’s mate in chess: you think you have mounted a powerful attack against your opponent’s king, but the next thing you know, you have been checkmated. Other proofs sent in by amateurs are correct, but they are merely rediscoveries of long-known results.

  What Hardy had before him that morning looked like real mathematics. But that didn’t mean it was necessarily the real deal. Was it fraudulent? he asked himself. Had this unknown Indian with the careful schoolboy handwriting that always missed crossing his t’s perhaps copied the work of a mathematician from an obscure journal and was trying to pass it off as his own? Or was the work perhaps genuinely his own but of no real value? Looking at the formulas more carefully, Hardy recognized some of the results as mathematical derivations that had been obtained by others and were well known. Others made no sense to him at all. Yet others were so fantastic that they had to be the work of eithe
r a crank or a genius. It would just have to wait. He put the letter aside, picked up his newspaper, and continued with his breakfast, planning to look at it more carefully that evening.

  After the morning’s regimen of four hours of mathematical research came lunch, then perhaps a game of tennis and dressing for dinner. That evening, Hardy again studied the letter from India. He became more and more intrigued. As he reread the pages, he realized that two results that had at first looked like nonsense were somehow related to hypergeometric series and continued fractions, which had been previously studied by Euler and Gauss.

  Continued fractions are numbers that are described by an iterative process of division. For example, we can express the fraction 7/10 as a continued fraction thus:

  Such a finite continued fraction always represents a rational number. Things get interesting when irrational numbers are developed into infinite continued fractions. Let us take as an example a famous irrational number, the golden ratio, denoted by the Greek letter phi,

  though any old square root would have done almost as well. Pythagoras, Euclid, and Kepler, among many other distinguished scientists through the ages, have been fascinated by this number. It is called golden because as a proportion, it is pleasing to the eye and has been used extensively in works of art. It also appears in nature, in such diverse places as the arrangement of leaves and branches of plants, the geometry of crystals, and the structure of DNA.

  The golden ratio also appears in many of mankind’s most beautiful creations. Some great works of architecture and art, such as the Parthenon, the pyramids of Giza, and the Mona Lisa, make use of the pleasing proportions of the golden ratio. Salvador Dalí, the famous twentieth-century Spanish surrealist, explicitly referenced the golden ratio in his masterpiece The Sacrament of the Last Supper, which depicts Jesus and his disciples seated below a dodecahedron, a geometric figure whose proportions define the golden ratio.

  Salvador Dalí’s The Sacrament of the Last Supper (courtesy of the National Gallery of Art, Washington, D.C.)

  As a purely mathematical object, the golden ratio has many faces. It is, for example, the limit of the ratios of successive terms of the famous Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … (each term is the sum of the previous two terms), namely the ratios 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55, … . The golden ratio was described by Euclid around 300 b.c.e. by “extreme and mean ratios,” which leads to the relationship

  whose positive solution is

  Despite these elegant descriptions, the infinite decimal expansion

  exhibits no apparent pattern. However, with some mathematical sleight of hand, a beautiful pattern can be made to emerge. Using repeatedly the relation

  we can develop ϕ into an infinite continued fraction:

  and so on ad infinitum. In his first letter to Hardy, Ramanujan indicated that he had found a theory that generalizes this well-known classical fact about the golden ratio. He claimed that he knew how to evaluate a much more general continued fraction, one containing a variable quantity that in theory can represent infinitely many continued fractions at once. Such continued fractions were well known to mathematicians, having been discovered in 1896 by the British mathematician Leonard Rogers. Thanks to the variable, the Rogers–Ramanujan continued fraction, as it is known today, generalizes the golden ratio, which is the special case in which the variable assumes the value 1.

  Ramanujan offered two examples, with the variable equal to and , where e = 2.718281… is Euler’s number. He claimed that the two resulting continued fractions are in fact equal to simple expressions involving the golden ratio. Immediately following these two formulas in the letter, Ramanujan dropped a bombshell. He made the astonishing claim that he had a general method for computing these continued fractions explicitly for a large class of values of the variable. If that were true, then Ramanujan possessed a theory unlike anything anyone had seen before.

  Hardy was amazed. Of all the values to choose from, why did Ramanujan choose the numbers and ? How did he then figure out their corresponding continued fractions? Hardy didn’t have a clue, prompting him later to remark,They defeated me completely, I had never seen anything in the least like this before … A single look at them is enough to show they could only be written down by a mathematician of the highest class. They must be true because no one would have the imagination to invent them.

  Hardy was not alone in his admiration. These identities have fascinated scores of mathematicians, serving as the topic of hundreds of research papers over the past century. The problem of discovering their source, a general theory that explains them as part of an infinite framework of identities, would become one of my main goals as a mathematician.

  Was this a startling new result or a bunch of nonsense? Hardy decided that he had better show the letter to his colleague Littlewood. Littlewood concurred that this curious letter seemed to contain some quite amazing results. The two mathematicians pored over the theorems and formulas in the letter for some time. They were captivated. They met a few more times, trying to make sense of the theorems and series, much of it written in nonstandard notation.

  As the days passed, and Hardy and Littlewood, and others to whom Hardy showed the letter, analyzed its contents, Hardy came to the conclusion that they were in the presence of something most unusual. First of all, the letter contained some mathematical results that he and Littlewood had seen before—albeit written in what seemed like a different language, because of the notation—but which, they decided, Ramanujan had likely not seen elsewhere. He had apparently rediscovered published results that had seen little circulation.

  Other results were completely new, and they were able to prove some of them, though it took considerable effort. Some assertions were incorrect, but such mistakes were rare, and even the erroneous results were clever, obtained through an interesting mathematical process. It appeared that Ramanujan had developed his own unique methods.

  It was clear that Ramanujan had discovered mathematical gems beyond the imagination, and Hardy was not going to let those treasures go to waste. But first, he wanted to see how Ramanujan had justified his results, for Hardy was a strong advocate of mathematical rigor. On February 8, he wrote to Ramanujan expressing his interest, but asking him kindly to supply proofs of his statements:I was exceedingly interested by your letter and by the theorems … You will however understand that before I can judge properly of the value of what you have done, it is essential that I should see proofs of some of your assertions.

  © Springer International Publishing Switzerland 2016

  Ken Ono and Amir D. AczelMy Search for Ramanujan10.1007/978-3-319-25568-2_14

  14. Permission from the Goddess

  Ken Ono1 and Amir D. Aczel2

  (1)Department of Mathematics and Computer Science, Emory University, Atlanta, GA, USA

  (2)Center for Philosophy & History of Science, Boston University, Boston, MA, USA

  Ramanujan was elated finally to receive a positive response from England, and he promptly replied to Hardy: “I have found a friend in you, who views my labours sympathetically.” But instead of proofs of the sometimes audacious statements he was making about infinite series, continued fractions, and integrals, Ramanujan sent Hardy more theorems. Hardy received these with disbelief. How could one young man, in a country so distant from the world centers of mathematics, come up with so many ingenious mathematical results? It was a mystery. But he persisted in requesting proofs. And Ramanujan persisted in not supplying them—instead sending even more results.

  Hardy began to develop a suspicion. Perhaps the Indian was worried that the English mathematician—whom he knew not at all—would steal his work and publish it as his own. And so Hardy wrote again, explaining that Ramanujan now possessed letters from him, which—if Hardy tried to pass his work off as his own—could be used to incriminate him in a clear case of academic theft. Perhaps this suspicion about a suspicion caused a cooling off of the epistolar
y relationship that had developed between them. For a while, there were no more letters.

  Hardy had to find out where these claims and formulas—marvelous, mysterious, beguiling—came from. Were they true, these mathematical treasures? And if so, why were they true? And how had this stranger on the other side of the world obtained them? Ramanujan would have to come to Cambridge.

  One of Hardy’s youngest colleagues at Trinity was Eric H. Neville, a man two years younger than Ramanujan. Neville was on his way to India for a research and lecturing project in his area of differential geometry. Hardy asked to speak with him before he embarked for India. Would he be willing, while in India, to travel to Madras, and through academic connections there, try to meet this mysterious young man, a certain Ramanujan? Then, Hardy suggested, if the meeting was satisfactory, he should be convinced to travel to England.

  Neville arrived in India, and after some time made it to Madras, where he met Ramanujan. The two men—perhaps because they were so close in age—got on very well. In fact, they would become lifelong friends. While in Madras, Neville began gently to coax the reluctant Ramanujan to go to England, where his gifts could be further developed, where he would be appreciated as a mathematician, and where he could work more freely without worrying about a livelihood. Neville reported to Hardy about his modest progress in convincing Ramanujan to think about coming to Cambridge, and Hardy then wrote an official letter of invitation.