My Search for Ramanujan

by Ken Ono

  I even held, in my own hands, Ramanujan’s notebooks, the ones that have been central to my mathematical career. As I held them, I was thinking, “Oh my God! Is this really okay? Have I earned the right to hold these sacred volumes?” We spent hours at the University of Madras, studying Ramanujan’s notebooks. I could have spent days examining its pages. Holding the notebooks, the source of so many treasures, was a spiritual experience that I will never forget. It is one of the highlights of my life as a mathematician.

  Perusing Ramanujan’s notebooks with Raghuram (photo courtesy of Emory University)

  It was clear that Ramanujan had edited these notebooks carefully—they clearly did not contain the raw results as he first obtained them. His mathematical formulas and equations are carefully written, on neat pages, usually only on the right side of the page, although the left side would later be used as scrap paper for calculations. Ramanujan wrote very carefully in green ink, clearly and legibly.

  Ramanujan’s notebooks (published after his death and edited by Bruce C. Berndt) contain some work he did he while still a student at Town High in Kumbakonam. Flipping through his first notebook, I immediately came across some of that work, his results on magic squares, which are square arrays of numbers such that the sum of the numbers in each row and column and on the main diagonals, and often in subsquares within the main body of the square, add up to the same number.

  Magic squares with three rows and columns, called 3 by 3 (or 3 × 3) squares, have been known since early antiquity. Chinese writings from about 650 b.c.e., the Lo Shu, or “Scroll of the River Lo,” contain a 3 × 3 magic square, an arrangement of the numbers from 1 to 9:

  Note that the sum of every row, of every column, and of the two main diagonals is 15, and that is what characterizes a “regular” 3 × 3 magic square. For a 4 × 4 magic square, containing the numbers from 1 to 16, the sums must be 34. In fact, in the tenth-century temple complex in north India called Khajuraho, there is an interesting 4 × 4 magic square, attesting to the fascination in ancient India for such objects. The fact that this square is in a temple suggests some religious or spiritual connection between mathematics and the gods.

  I stared in awe at the huge 8 × 8 square Ramanujan had constructed and recorded in his first notebook. He also discussed some properties he saw in magic squares, for example, that the middle row, middle column, and each diagonal in a 3 × 3 square form an arithmetic progression. Arithmetic progressions are sequences in which the adjacent numbers share a common difference. In the 3 × 3 square above, the arithmetic progressions in question are (3, 5, 7), (9, 5, 1), (4, 5, 6), and (8, 5, 2).

  The documentary provided me with a second pilgrimage to pay homage to Ramanujan. This time, I had the chance to share it with Indian filmmakers and mathematicians who would be telling the story to the public. This trip, with Kudhyadi, Raghuram, and my new friend Venkateswaran Thathamangalam Viswanathan, filled in many of my cultural gaps in the Ramanujan story.

  Face to Face with Ramanujan

  If anyone were to ask me what one person I would want to meet in all of history, my answer, as the reader may easily guess, would be Srinivasa Ramanujan. Amazingly, my wish came true, in a way.

  In the last week of July 2014, I received an unexpected email from film director Matthew Brown. I had heard eight years earlier that Matt had written a screenplay for a film on Ramanujan based on Robert Kanigel’s book The Man Who Knew Infinity. I hadn’t heard anything further about it in years, and so I naturally assumed that the project had been scrapped.

  I was delighted to learn from Matt that the film was well into preproduction. I was already excited by the promise of the forthcoming “mathy” biopics The Theory of Everything, about Stephen Hawking, and The Imitation Game, about Alan Turing. Those films would soon earn wide critical acclaim, including multiple Oscar nominations. Both films would go on to win Academy Awards. Perhaps this film on Ramanujan would be the next British math prestige film.

  Dev Patel, the twenty-four-year-old star of the hit Slumdog Millionaire, was cast to play Ramanujan, and Academy Award winner Jeremy Irons was cast to play Hardy.

  The Pressman Film Company, producers of over eighty films including Conan the Barbarian, The Crow, and Wall Street, was producing the film at Pinewood Studios, in London. Filming was set to begin early in August.

  Matt’s email was short. He wanted to skype the next morning to discuss the film. He mentioned something about technical assistance for the art department. I had no idea that this would lead to a whirlwind opportunity of a lifetime, the closest I would ever get to meeting Ramanujan face-to-face.

  The next morning, I skyped with Matt and Liz Colbert, an Irish graphic artist well known for her work on the TV show Game of Thrones and the blockbuster film Sherlock Holmes, which starred Robert Downey Jr. Liz was charged with producing the mathematical props for the film. She needed help in identifying the documents to be used, which would have to be painstakingly reproduced by hand. My job was to choose these documents, including some of Ramanujan’s letters to Hardy and select pages from the notebooks.

  Our skype session extended well beyond the scheduled thirty minutes. There was simply too much to discuss, and much work to do. I could tell that the art department was relieved to know that I knew my stuff. I knew the documents by heart, and I knew the story. Within minutes, I was the man in charge of the math. I was thrilled that Matt had decided to pay attention to such details. There was to be no fudging. He wanted to be true to the story, and that included exceptional attention to detail regarding the mathematics in the film.

  Matt thanked me for my offer of help, clearly relieved that getting the math right was no longer a worry. He had no idea that I would have dropped everything to work on the film.

  I wanted to know about the last-minute tasks that had to be completed before filming could begin, and I learned about the frenetic last days of preproduction. Matt had to finalize schedules, oversee the production of props, visit locations, and he had yet to finish casting. Among his concerns, he mentioned a little detail that the research department had been struggling to solve. They had been unable to find Janaki’s autograph. Even though it probably would not appear in the film, Matt wanted it just in case. Did I have any leads?

  I proudly exclaimed that I didn’t need any leads, because I had Janaki’s autograph. In fact, her autograph had been in my possession for fifteen years, and in my father’s for the fifteen years before that, since that day in 1984 when her letter to my father arrived in our mailbox in Lutherville, Maryland. Matt and Liz looked at each other and laughed in surprise at their good luck. Luck? I wasn’t so sure.

  A few hours later, Matt emailed me again with a handsome offer that I couldn’t refuse. So I didn’t, and three days later, I found myself on a flight to London. I would end up spending several weeks working on the film. At first, my job was to help the art department with the formulas. I had no idea that I would end up doing much more. In fact, I would ultimately be named an associate producer.

  The Pressman Production Company arranged a luxurious hotel room for me near Paddington Station, the location being based on my requirement of proximity to a competition swimming pool. I was training for the International Triathlon Union’s Cross Triathlon World Championships, which would be held in Zittau, Germany, at the end of August.

  Preproduction at Pinewood Studios, which was also filming Star Wars: Episode VII, was frantic. The art department was busily producing props, including copies of Ramanujan’s letters and notebooks. Liz and I carefully went over the documents that she needed scene by scene.

  Three days before filming was to begin, Jeremy Irons and Dev Patel arrived at the studio for rehearsals. I had hoped to get a glimpse of them working with Matt. I was working with Liz in her office when a production assistant rushed in with the message that the principals wanted to see me.

  A few minutes later, I was in a quiet room with Matt, Jeremy Irons, and Dev Patel, secluded from the rest of the prep
roduction frenzy. Matt introduced me, and I nervously shook their hands. Matt handed me a script and said that we were going to rehearse. Oh my God! Oh my God! I couldn’t believe it. I was going to get more than a glimpse!

  Jeremy and Dev had asked for me because they wanted to know how a mathematician thinks. No true actor is content merely to deliver his lines. How did Hardy and Ramanujan think about mathematics? How did they speak about it? Write about it? What sort of gestures would they have used? They explained to me that what they wanted from me was something like what actors get from a dialect coach. They couldn’t slip into their roles without an expert on hand explaining how these men would have thought and spoken.

  The four of us sat in a circle in a second-floor room at Pinewood Studios. This was the place that had made the James Bond films, the Harry Potter films, the Captain America films, and the Sherlock Holmes films. Oh my God, and I was there with a job to do! This was more than a tour of the Wonka chocolate factory. I was going to help make the chocolate!

  Those rehearsals were amazing in many different ways. As a fan of film, it was my chance to observe firsthand some of the world’s finest actors at work. I had no idea how difficult it is to be a serious actor. Our work on the scenes followed a standard pattern. Matt would begin by giving the outline of each scene and its role in the film. Then the actors would read the lines out loud. The three of them would then discuss what they had to do to make the scene work. Perfecting a scene involved reworking lines, working out facial expressions and body language, and discussing the props, all in a quest for authenticity.

  With Jeremy Irons on the set (photo by Sam Pressman)

  They all had lots of questions for me. By the end of the first hour, I understood what was required of me, and I was comfortable enough to speak up and offer suggestions and comments. They trusted my opinion and judgment as the math consultant. All along, I was thinking that my passion for Ramanujan and his mathematics was about to make the world stage, performed by the world’s best. Unbelievable!

  Going over the script with Dev and Jeremy was beyond a dream come true. It wasn’t even something I would ever have dreamt of wishing for. Although I doubt that Hardy ever called anyone “babycakes,” as Jeremy did me, I will forever cherish these rehearsals as the closest I will ever come to being face-to-face with Hardy and Ramanujan.

  To celebrate the end of preproduction, Edward Pressman hosted a small private party at a fancy London restaurant. To my surprise, he asked me to say a few words about Ramanujan and his mathematics. Although I am accustomed to giving lectures about Ramanujan, this short speech was something special. With a glass of champagne in hand, I did my best to speak eloquently about Ramanujan, the man I had been searching for most of my life. And there was Dev Patel, the man who would play Ramanujan, seated directly across from me.

  Preproduction party (left to right: Edward Pressman, Jeremy Irons, Dev Patel, Ken Ono, Matt Brown, Sorel Carradine) (photo by Sam Pressman)

  My Search Goes On

  Building on the success I enjoyed in seeking the hidden meaning in Ramanujan’s writings—his claim that there are no other simple properties for partitions other than the ones he had found, his search for a simple law that determines the odd numbers not of the form x 2 + y 2 + 10z 2, I decided to study his works in detail on the belief that Ramanujan, speaking beyond the grave, has vouchsafed hints of beautiful theories to the mathematicians of today. I hear him speaking to me when I read his papers. And if I read them with care, I might be lucky enough to figure out what he had in mind.

  I have spent the last five years as a professor at Emory University, working on some of Ramanujan’s mathematics and its implications for contemporary questions. Jan Bruinier and I revisited the partition numbers and the work of Hardy and Ramanujan that gave the approximate formula for those numbers described below in the section on Euler’s partition numbers. In the 1930s, building on the Hardy–Ramanujan formula, which gives good approximations but never the exact values, Hans Rademacher obtained an exact formula. However, his formula had a catch. His description of the partition numbers was as a convergent infinite sum. (Since a partition number is an integer, you could use Rademacher’s formula to compute individual partition numbers: you know that you have arrived at the correct value after adding a finite number of terms when you can estimate the sum of the infinite number of remaining terms as being less than a small fraction.) We aimed to find a different conceptual formula—one that involved only finitely many terms.

  We reformulated the problem in terms of concepts that we had already been using on related problems. By modifying work that Jan had done with Jens Funke, we produced a formula that gives the exact values of the partition numbers as a finite sum of algebraic numbers (these are relatively well behaved rational or irrational numbers) that are themselves values of a single special function. And in joint work with Drew Sutherland of MIT, we were able to express our formula in a way that could be, and now has been, implemented on a computer.

  I have thought very deeply about the mathematics in Ramanujan’s first letter to Hardy. The first challenge of that letter was to understand how Ramanujan came up with his R(q) continued fraction, the one that generalizes the golden ratio. Ole Warnaar, a well-known mathematician at the University of Queensland, had been working on this problem too.

  I learned two years ago that Ole had made a huge breakthrough concerning certain hypergeometric series transformation laws, results that experts believed could hold the key to finding more identities of the type that Rogers and Ramanujan had discovered a century earlier. Ole had found infinite families of power series identities that included the special continued fractions that Ramanujan discussed in his first letter to Hardy.

  I immediately contacted Ole by email, and I suggested that we join forces. Together with my PhD student Michael Griffin, we made use of Ole’s beautiful formulas to establish a framework in which to place the Rogers–Ramanujan identities, which we then put to good use to obtain infinitely many generalizations of the golden ratio, algebraic units that are the values of the functions that Ole had discovered. What Ramanujan had offered in his first letter to Hardy turned out to be the first example of the functions we now understood.

  Discover magazine ranked our accomplishment fifteenth among the top hundred stories in science of 2014. The editors conducted a “People’s Choice Award,” and our work on Ramanujan’s first letter came in second.

  George Andrews, Michael Griffin, Ken Ono, Ole Warnaar, Jim Lepowsky

  Like his first letter to Hardy, Ramanujan’s last also speaks to the mathematicians and physicists of today. In that letter, he described his enigmatic mock theta functions, which have been studied by the Japanese physicists Tohru Eguchi, Hirosi Ooguri, and Yuji Tachikawa and the Canadian mathematician Terry Gannon. Building on their work, Miranda Cheng (University of Amsterdam), John Duncan (Case Western), and Jeff Harvey (UChicago) formulated a conjecture in 2010 that predicted a very deep and precise relationship between Ramanujan’s mock theta functions—the stuff of Ramanujan’s deathbed letter to Hardy—and the hottest item in theoretical physics today: string theory.

  With Ramanujan’s last letter to Hardy in 2013 at Trinity College

  Their conjecture, which they called the umbral moonshine conjecture, was formulated to place Berkeley mathematician Richard Borcherds’s 1992 proof of the so-called moonshine conjecture, for which he was awarded the Fields Medal, in a wider context. If correct, Borcherds’s work would be the first of many different “moonshine” theories. Mathematical physicists were now realizing that such theories are important in string theory, which aims to answer a fundamental question: what is the universe made of?

  In joint work with John Duncan and my PhD student Michael Griffin, we proved the umbral moonshine conjecture, a result that now places the mathematics of Ramanujan’s deathbed letter front and center in cutting-edge mathematical physics. We had proved that the functions he conjured in the last months of his life encode astoni
shing symmetries in the world of mathematics, and experts now predict that his functions will be put to good use in the study of black holes, quantum gravity, and the theory of everything. Discover magazine has informed us that our accomplishment will be among the top 100 stories in science of 2015.

  John Duncan, Ken Ono, Michael Griffin

  The search for Ramanujan shall go on. His words still speak to us.

  My mathematical search for Ramanujan is a never-ending story. As I continue my quest, I find again and again that I have overestimated my ability to understand the full meaning in Ramanujan’s notebooks, letters, and papers. Ramanujan’s legacy is inexhaustible.

  Afterword

  Two Questions

  I am often asked about Ramanujan and his story. Was Hardy the best mentor for Ramanujan? Was Ramanujan the greatest mathematician of his time—or perhaps of all time?

  Was Hardy the best mentor for Ramanujan? This is a difficult question to answer. People have raised this question for a variety of reasons. Hardy was not an expert on theta functions and modular forms, areas in which one arguably finds Ramanujan’s most important contributions. Would mathematics have advanced further had he been mentored by an expert in those fields? Some say that Hardy selfishly made use of Ramanujan’s intellect instead of helping him to become a more professional mathematician. Would Ramanujan have been a more important mathematician had he been mentored by someone who did not need him as a collaborator?

  Although these are legitimate questions, the point is moot. We cannot travel back in time and change history. But if we must wonder, here are my thoughts. In India, Ramanujan worked in a vacuum; there was no one to nurture his talent. Hardy recognized Ramanujan’s ability when others ignored him, and he brought him to England and helped him make his mathematics understandable and presentable to the world. Had this never happened, then Ramanujan would likely have disappeared without a trace. Had this never happened, I would never have happened. I will therefore always hold Hardy in very high regard. Opinions vary on whether Hardy could have taken steps that would have saved Ramanujan’s life. I am not qualified to speculate, and I don’t wish to speculate.