Honoring a Gift from Kumbakonam Ken Ono oday was an absolutely This adventure was a pilgrimage to glorious day in Madison, pay homage to Srinivasa Ramanu- Wisconsin. It is Christmas jan, the Indian legend whose con- 2005, and everyone in the gruences, formulas, and identities Thouse is asleep after a long have inspired much of my own work. day of enjoying family, opening pre- This fulfilled a personal journey, one sents, and eating enormous portions of with an unlikely beginning in 1984. mashed potatoes and yule log cake. Yet powerful images keep me The Story of Ramanujan awake. Ramanujan was born on December Thirty-six hours ago I returned 22, 1887, in Erode, a small town from a six-day whirlwind jour- about 250 miles southwest of Chen- ney to a far-off place. I spent nai (formerly known as Madras). forty hours on airplanes, and I He was a Brahmin, a member of endured fourteen hours in cars India’s priestly caste, and as a con- dodging bicycles, rickshaws, sequence he lived his life as a cows, goats, and masses of peo- strict vegetarian. ple on roads severely damaged When Ramanujan was one year by recent flooding. These floods old, he moved to Kumbakonam, a would be blamed1 for at least forty-two small town about 170 miles south of deaths. Despite these hardships and Chennai, where his father Srinivasa bad luck, this adventure exceeded my Bust of Srinivasa was a cloth merchant’s clerk. Kum- lofty expectations. Ramanujan by artist Paul bakonam, which is situated on the I ostensibly travelled to Kum- Granlund. banks of the sacred Kaveri River, bakonam with the purpose of giving a was (and remains today) a cos- lecture on mock theta functions and Maass forms mopolitan center of the rural Indian district of at the International Conference on Number Theory Tanjore in the state of Tamil Nadu. Thanks to the and Mathematical Physics at SASTRA University. I area’s rich soil and tropical climate, rice and sugar could have offered other worthy pretexts: I wanted cane crops thrive. In Ramanujan’s day, Kum- to see my student Karl Mahlburg give his first ple- bakonam had a population of fifty thousand. nary lecture. I wanted to applaud my friends Man- Kumbakonam is one of India’s sacred Hindu jul Bhargava and Kannan Soundararajan (he goes towns. It boasts seventeen Hindu temples (eleven by Sound) as they won a prestigious prize. However, honoring the Hindu god Lord Siva, and six honor- my primary reason was personal, not professional. ing the god Lord Vishnu). The town is perhaps most well-known for its Mahamaham Festival, which Ken Ono is the Solle P. and Margaret Manasse Professor is held every twelve lunar years when the Sun en- of Letters and Science at the University of Wisconsin, Madi- ters the constellation of Aquarius and Jupiter en- son. His email address is [email protected]. ters Leo. Nearly one million Hindu pilgrims de- 1This was reported in The Hindu on December 20, 2005. scend on Kumbakonam for the festival. In a ritual 640 NOTICES OF THE AMS VOLUME 53, NUMBER 6 In July 1909, Ramanujan married nine-year-old meant to absolve sins, pilgrims bathe in the S. Janaki Ammal; it was an arranged marriage. After Mahamaham tank, which symbolizes the waters of a short stay with Ramanujan and his family, Janaki India’s holy rivers. returned to her home to learn domestic skills and As a young boy, Ramanujan was a stellar student. pass time until she reached puberty. Ramanujan He entered Town High School in 1898, and he moved to Madras in 1911 and Janaki joined him in would go on to win many awards there. He was a 1912 to begin their married life. To support them, strong student in all subjects, and he stood out as Ramanujan took a post as a clerk in the account- the school’s best math student. His life took a dra- ing department of the Madras Port Trust. matic turn when a friend loaned him the Govern- Ramanujan continued his research in near iso- ment College library’s copy of G. S. Carr’s Synop- lation. His job at the Port Trust provided a salary sis of Elementary Results in Pure Mathematics. G. H. Hardy, the celebrated Cambridge professor, and left time for mathematics. Despite these cir- later described (see page 3 of [17]) the book as cumstances, his frustration mounted. Although some Indian patrons acknowledged his genius, he …the “synopsis” it professes to be. It was unable to find suitable mentors. Indian math- contains enunciations of 6,165 theo- ematicians did not understand his work. rems, systematically and quite scien- After years of such frustration, Ramanujan tifically arranged, with proofs which are boldly wrote distinguished English mathemati- often little more than cross-references… cians. He first wrote H. F. Baker, and then E. W. Hob- Ramanujan became addicted to mathematics son, both times without success. His letters con- research, and he recorded his findings in note- sisted mostly of bare statements of formal books, imitating Carr’s format. He typically offered identities, recorded without any indication of proof. no proofs of any kind. Based on his education, he Due to his lack of formal training, he claimed some presumably did not understand the obligation known results as his own, and he offered others, mathematicians have for justifying their claims such as his work on prime numbers, which were with proofs. plainly false. In this regard, Hardy would later write Thanks to his exemplary performance at Town (see page xxiv of [16]): High School, Ramanujan won a scholarship to Gov- ernment College. However, by the time he enrolled Ramanujan’s theory of primes was vi- there in 1904, his addiction to mathematics made tiated by his ignorance of the theory of it impossible for him to focus on schoolwork. He a complex variable. It was (so to say) unceremoniously flunked out. He would later get what the theory might be if the Zeta- a second chance, a scholarship to attend function had no complex zeroes. …Ra- Pachaiyappa’s College in Madras. However, math- manujan’s Indian work on primes, and ematics again kept him from his schoolwork, and on all the allied problems of the theory, he flunked out a second time. was definitely wrong. By 1907, the gifted Ramanujan was an academic failure. There was no room for him in India’s sys- Ramanujan’s work on Bernoulli numbers, which tem of higher education. Despite his failures, his he presumably included in his letters, also includes friends and parents supported him. They must an incredible mistake involving explicit numbers. have recognized his genius, for they allowed him The Bernoulli numbers [23] are the rational num- to work on mathematics unabated. Vivid accounts 2 bers B2 = 1/6,B4 = 1/30,... defined by portray Ramanujan hunched over his slate on the B B B porch of his house and in the halls of Sarangapani x cot x = 1 − 2 (2x)2 − 4 (2x)4 − 6 (2x)6 −··· . Temple, working feverishly. 2! 4! 6! ….Ramanujan would sit working on the Ramanujan falsely conjectured (see equation (14) pial (porch) of his house on Sarangapani of [23]) that if n is a positive even number, then the Street, legs pulled into his body, a large numerator of Bn/n, when written in lowest terms, slate spread across his lap, madly scrib- is prime.3 This conjecture is false, as is plainly bling, …When he figured something out, seen by he sometimes seemed to talk to himself, B 174611 283 × 617 smile, and shake his head with plea- 20 = = . sure. 20 6600 23 × 3 × 52 × 11 R. Kanigel (see page 67 of [20]) 2This is a slight departure from the modern definition of the Bernoulli numbers b2n. These numbers are related by n+1 It is said (for example, [3, 20]) that Ramanujan the relation B2n =(−1) b2n . believed that his findings were divine, told to him 3Ramanujan obviously considered 1 to be a prime for in dreams by Namagiri, the goddess of Namakkal. this conjecture. JUNE/JULY 2006 NOTICES OF THE AMS 641 In fact, among the even numbers n less than 2000, Newton. News of his election spread quickly, and Ramanujan’s conjecture holds only for the twenty in India he was hailed as a national hero. numbers Ramanujan grew ill towards the end of his stay 2, 4, 6, 8, 10, 12, 14, 16, 18, 26, 34, 36, in England. One of the main reasons for his de- clining health was malnutrition. He was a vegetar- 38, 42, 74, 114, 118, 396, 674, 1870. ian living in World War I England, a time when al- In view of these facts, it is not surprising that Baker most no one else was a vegetarian. Ramanujan also and Hobson dismissed him as a crank. struggled with the severe change in climate; he Then on January 16, 1913, Ramanujan wrote was not accustomed to English weather. He did not G. H. Hardy, a thirty-five year old analyst and num- have (or did not wear) appropriate clothes to pro- ber theorist at Cambridge University. With his let- tect himself from the elements. These conditions ter he included nine pages of mathematical scrawl. took their toll, and Ramanujan became gravely ill. C. P. Snow elegantly recounted (see pages 30-33 of He was diagnosed with tuberculosis. More recently, [18]) Hardy’s reaction to the letter: hepatic amoebiasis [4, 29], a parasitic infection of One morning in 1913, he (Hardy) found, the liver, has been suggested as the true cause of among the letters on his breakfast table, his illness.