My Search for Ramanujan

by Ken Ono

  The ordinary Riemann hypothesis is one of the most important open problems in mathematics. It involves a certain conjectured property of a certain function of a complex variable. Its truth would resolve many unanswered questions. For example, mathematicians would have a much clearer understanding of prime numbers if the Riemann hypothesis were confirmed to be true. The generalized Riemann hypothesis is a natural generalization of the Riemann hypothesis.

  Using a long and complicated argument, we finally found a way to show that the truth of the generalized Riemann hypothesis implies that every odd number greater than 2719 can be written as for some integers x, y, and z. The fact that almost every mathematician believes in the truth of the generalized Riemann hypothesis and the fact that every odd number greater than 2719 up to a very large number can be represented by Ramanujan’s quadratic form convinced us that we had found the law. But although the law is simple enough to state, it thus far defies a definitive proof. To be sure, if someone manages to prove the generalized Riemann hypothesis, then our conditional proof will at once become a genuine proof. But the generalized Riemann hypothesis is arguably one of the most difficult open problems in mathematics. So Ramanujan was right that the odd numbers do not obey a simple law, in the sense that they are constrained by one of the most difficult unsolved problems in mathematics.

  I had no idea that I would see the number 2719 again ten years later, etched on a wall in the very spot where Ramanujan performed some of his first calculations.

  Thanks to my two years in Princeton, I was able to complete a large body of work, and I was rewarded for my efforts with a tenure-track assistant professorship at Penn State University. George Andrews, one of the mathematicians I admired from the BBC documentary and Robert Kanigel’s book The Man Who Knew Infinity, had recommended me for the position. My work on Ramanujan’s mathematics, namely my work on Subbarao’s conjecture for the partition numbers and my work with Sound on Ramanujan’s quadratic form, were the reasons for my success.

  © Springer International Publishing Switzerland 2016

  Ken Ono and Amir D. AczelMy Search for Ramanujan10.1007/978-3-319-25568-2_31

  31. Bittersweet Reunion

  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

  Princeton, New Jersey (1997)

  Shortly before we moved to Penn State, my parents made the drive from Baltimore, where my father was still teaching at Johns Hopkins, for a weekend visit. I had arranged for them to be reunited with André Weil, who was now ninety-one. My parents had not seen Weil since the 1970s, and they didn’t know what to expect. They were nervous. However, the reunion was important to them. They needed to thank him for his generosity, for having made their lives possible.

  Watching my parents walk the grounds of the Institute and hearing them retell the story of how André Weil rescued them as a young starving Japanese couple in 1955 was profoundly moving. Weil had invited my father to the Institute, and the hallowed grounds now represented the beginning of everything that was good in their lives. They excitedly pointed out where they used to go for walks, and they jokingly pointed out the “Tamagawa tree,” a tree on Einstein Drive that Tsuneo Tamagawa, their friend and future Yale professor, had rammed with his car while learning to parallel park.

  Sitting in the Institute’s Fuld Hall waiting for Weil to arrive, my father talked about the Tokyo–Nikko conference. I knew only a small part of the story, the bits that most professional number theorists know. But there was more that my father wanted to tell me.

  At that conference, Weil gave an impromptu after-dinner talk that wasn’t part of the official program. He wanted to inspire the young Japanese mathematicians by telling a story that had inspired him when he had been uncertain of his future in mathematics.

  Weil had a fascinating life, in a way reflecting that of my father. Although he was born into a prominent Alsatian Jewish family and grew up in privilege in Paris, he had struggled early in his career.

  As a young mathematician, Weil traveled to India and soaked up the culture—not many Western mathematicians were doing that in the early twentieth century—and throughout his life, he was influenced by Hindu thought. Weil was traveling in Scandinavia in 1939 and was in Finland when war broke out between Finland and Russia. He was mistakenly arrested by the Finns for spying. He was soon released, but on his return to France, he was arrested for failing to report for military duty, and he spent time in prison. In 1941, he managed to sail with his family from Marseille to New York, and for several years he was a mathematical nomad, struggling in his own kind of purgatory for a mathematician of his stature.

  Weil was saddled with a heavy teaching load at Lehigh University for two years, followed by two years in a low-paying position at the Universidade de São Paulo, in Brazil. His troubles were somewhat reminiscent of Ramanujan’s own struggles. Finally, the importance of his work was fully recognized, and he was suitably rewarded with a faculty position at UChicago in 1947.

  Inspired by the purpose of the symposium—the promotion of reconciliation and world peace in the name of science—Weil felt the need to tell a story that had inspired him when he needed hope. Dressed in a white shirt and light khaki pants, Weil addressed the Japanese mathematicians, almost all in dress shirts, neckties, and dark suits, politely sitting in formation at tables behind cardboard rectangles bearing their names.

  He spoke about a most enigmatic man, an untrained college dropout from faraway India, who had overcome unbelievable obstacles forty years earlier to startle a few of the world’s leading mathematicians with his creativity and imagination before his untimely death at the age of thirty-two. The young Japanese men had never heard of that man, Srinivasa Ramanujan, but by the end of the evening, they had all embraced a new hero. Ramanujan had inspired Weil, and that night, he inspired those young Japanese mathematicians, hungry in body and spirit, who needed hope and inspiration.

  André Weil lecturing at the 1955 Tokyo–Nikko conference

  Oh my God, it all made sense now!

  It was no wonder that my father had been so overcome by Janaki’s letter, though it was little more than a form letter that she had sent to dozens of mathematicians. I now understood that on that fateful day in 1984, Janaki’s letter had revived within my father memories of his own life’s pivotal moment. Ramanujan had been an inspiration to him, a mathematical hero who embodied everything admirable in an enquiring mind pitted against hardship. And he had learned about Ramanujan at the conference where he himself had been discovered by the great André Weil, setting in motion everything that had been good in his life. Janaki’s letter had carried my father to that moment in 1955 when it was revealed that all his efforts had not been in vain, that he, too, counted.

  The clouds parted; the sky was suffused with light; somewhere, an angelic choir was singing; the voices in my head fell silent. I saw before me my destiny and purpose. I was to be the next link in the chain. I was to lead a life that mirrored my father’s path. We had both earned positions at the Institute after overcoming formidable challenges. And it turned out that we had both benefited from strong mentors who helped pave our way. Had it not been for Weil’s generosity, what would have become of my parents? Had it not been for the strong support of men like Sally, Gordon, and Granville, and had it not been for Ramanujan and the “Ramanujan miracles” that seemed to occur when I needed them most, what would have become of me?

  When I offered the example of Ramanujan in a last-ditch effort to convince my parents to let me drop out of high school, I did so because I was grasping at straws. But in uttering the Open Sesame of Ramanujan’s name, I unwittingly did much more than secure my freedom. I offered my father a glimpse of the next chapter in an infinite story that continues to evoke awe and wonder.

  Janaki’s letter had reminded him that not all paths can be safely laid ou
t in advance. And that recollection allowed him to see me in a different light. Perhaps I could find my own way toward living the life that was meant for me, a life that made good use of my talents. It was that glimpse of possibility that convinced him to let me go. The enigmatic Ramanujan had helped make my father, and because of that, he was somehow helping to make something of me.

  The end to the weekend was bittersweet. The ninety-one-year-old Weil finally arrived. Although he had long since retired, he still spent much of his time at the Institute, sitting alone in a comfortable chair in the tea room of Fuld Hall. He rarely spoke to Institute members. He kept to himself, and he seemed perfectly content in doing so.

  Weil was now nearly deaf and blind. He was unable to focus his gaze in the direction of my parents as they addressed him. It was clear to me that Weil remembered my parents, and I am certain he understood something of what they were saying. However, all he could muster in return, in a soft melancholy voice, were the words, “I am so sorry Ono. I cannot hear nor see you.” And that is how it ended.

  In tears at the sight of this once powerful man, they left Weil alone in his comfortable chair, staring off into space. The man who had been instrumental in virtually everything important in our lives, who had been a towering figure in twentieth-century mathematics, had been reduced by age and infirmity to a mere shadow of his former self. Weil would pass away within the year.

  That weekend was the first time that I was able to offer my parents convincing evidence that I had achieved a level of success of nonzero measure. I had become an adult in their eyes and was no longer disparaged as a high-school dropout and unmotivated college student. It didn’t matter that I had quit the violin, nor that my complex-analysis professor felt that I had insufficient talent. It was of no importance that I had almost failed my algebra qualifying exam. That my first two hundred job applications had resulted in zero interviews was a thing of the past. Those events, which had been delicious nourishment for the negative voices in my head, had lost their hold. I was hosting my parents as a member of the Institute for Advanced Study, Einstein’s institute in the woods. I had reconnected them with Weil, allowing them to recall all the contingencies of their own journey from war-torn Japan to the Baltimore suburbs.

  That weekend marked a turning point in my relationship with my parents. They now understood that I was developing a strong reputation as a mathematician. They also enjoyed their role as grandparents to Aspen, even pushing her around the Institute grounds in a stroller, reliving the days thirty years earlier when I was the one being pushed. As if a wicked sorcerer’s spell had been lifted, my parents stopped worrying about me. It was a huge relief to me to realize that I had achieved a measure of professional success in their eyes. They stopped voicing their disapproval of me, no longer harping on my perceived inadequacies and failures.

  Since that moment, they haven’t missed holidays, birthdays, and anniversaries—all the celebrations that were absent in my previous life. I can now count on my mother to send cards for each of those occasions, just as I could count on her setting out my breakfast all those years ago. Years later, when they are seventy-eight and eighty-six, my parents will even travel to Atlanta to celebrate Aspen’s high-school graduation.

  © Springer International Publishing Switzerland 2016

  Ken Ono and Amir D. AczelMy Search for Ramanujan10.1007/978-3-319-25568-2_32

  32. I Count Now

  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

  State College and Washington, D.C. (1997–2000)

  Aspen, Erika, and I moved to State College, Pennsylvania, home of Penn State, in July 1997. To celebrate my appointment, George Andrews and I organized the conference “Topics in Number Theory in Honor of Basil Gordon and Sarvadaman Chowla.” The meeting was held from July 31 to August 3 at the Penn State Hotel. George and I wanted to honor the memory of Chowla, a longtime member of the Penn State faculty who had passed away in 1995, and we wanted to celebrate Gordon’s sixty-fifth birthday.

  It was a glorious meeting. Over 170 number theorists from all over the world attended the conference and joined our celebration. We heard about cutting-edge research from such stars as Henri Darmon (Princeton), Richard Stanley (MIT), and Trevor Wooley (University of Michigan), in addition to my friends Andrew Granville, Carl Pomerance, Chris Skinner, and Kannan (Sound) Soundararajan.

  I couldn’t think of a better way to celebrate Gordon’s birthday than to offer him this conference in his honor. Gordon had taught me how to love math for math’s sake. He had offered me strength when I was at my lowest point.

  I had prepared a moving speech for the conference banquet in which I would thank Gordon for everything he had done for me—showing me how to see beauty in mathematics, helping me to overcome my insecurities, and believing in me when I didn’t believe in myself. I wanted him to understand how important he had been in my life. I spent days working on that speech, discarding perhaps a dozen drafts. I agonized over the words. I had to get it right. Finally, I pared away all the high-sounding periphrasis and circumlocution and wrote a simple three-minute talk straight from my heart.

  It was the night of the banquet. When it was my turn to speak, I walked to the podium and unfolded the paper on which my speech was written. Standing before family, friends, and 170 fellow number theorists, I broke down and cried. I tried to find the strength to read my speech, but I was so overwhelmed by the emotions stirred up at the sight of its first words that I couldn’t even begin. When it became apparent that I was not going to regain my self-possession, that I was going to continue to stand there, choked up, unable to utter a word, Gordon walked up to the podium and embraced me. He didn’t need to hear my words. He knew, and everyone knew. I might have saved myself the trouble of writing a speech and simply borrowed that overused phrase, “Words cannot describe what Basil Gordon means to me.” I am crying as I write this.

  I handed the folded paper to Gordon. It began with the words, “I thank Basil Gordon for saving my life.”

  Basil Gordon passed away in 2012 after a long, happy, and fulfilled life. I was honored that his family asked me to speak at his memorial service. This time, in front of his family and former UCLA colleagues, I fought through tears, tears shared by everyone there, and I finally delivered my speech. I miss him deeply, and I thank God for sending me Basil Gordon.

  Shortly after the conference, I received an exciting offer from Bruce Berndt. Aware of my plan to search for Ramanujan the mathematician, with the idea of seeking the theories of which his claims offered enticing glimpses, Berndt asked me to help him edit one of Ramanujan’s unpublished manuscripts. The manuscript in question was an extensive collection of notes on Ramanujan’s tau-function and Euler’s partition numbers. We planned to publish a paper under Ramanujan’s name, to which we would add commentary for interested readers.

  I had already planned to study Ramanujan’s writings, looking for the clues that I felt had been placed there for me to discover. The proposed project with Berndt offered the perfect opportunity, and it involved one of Ramanujan’s unpublished manuscripts.

  Most of these unpublished notes involved ad hoc calculations with modular forms, and so Bruce thought of me as a natural candidate to help him with the task. I could provide a modern perspective, one based on the ideas of Deligne, Serre, and Swinnerton-Dyer. I had no idea that Bruce’s request would lead to insights that would give an enormous boost to my career.

  It was exciting to read Ramanujan’s unpublished notes. His 1919 observation, “It appears that there are no equally simple properties … involving primes other than these three,” had been haunting me for two years. The “simple properties” to which Ramanujan referred were his partition number patterns for the primes 5, 7, and 11. Did he know of properties for other primes? If so, were the patterns complicated? Apart from
my work on Subbarao’s conjecture, which concerned even and odd partition numbers, and work of A.O.L. Atkin from the 1960s, virtually nothing was known about these questions. It was from reading Ramanujan’s unpublished manuscript that I began finally to see what Ramanujan probably had in mind.

  My Penn State office was on the top floor of the McAllister Building, a former women’s dormitory that was apparently no longer fit to house undergraduates but was considered good enough for mathematicians. Built in 1904, the building was in desperate shape. I could hear squirrels rummaging in the attic above my office. The Internet went out several times during my first year: it seems that squirrels like the taste, or at least the mouthfeel, of cables. I had a window whose lock had been torn off years earlier. Fortunately, there was no way of opening it, since it was permanently sealed shut with layer upon layer of heavy white paint.

  I furnished my office at Penn State Salvage, a warehouse of castaway odds and ends scavenged from the dorms. I bought a stained freakish orange sofa for $20, and I placed it under the low ceiling that followed the gabled roofline of the building. It was on that sofa that I did much of my best work at Penn State, and it was there that I came across a few formulas related to partition numbers in Ramanujan’s unpublished manuscript that made no sense. I had been thinking about partition numbers for three years, and I became convinced in looking at his formulas that Ramanujan must have made some errors in his calculations. But I was unable to pinpoint them. I then decided that I was probably misunderstanding what he meant, which is common for anyone reading Ramanujan’s writings. I didn’t even see how to compute his expressions. His formulas simply looked too strange to be true.