SPIRIT OF RAMANUJA N xin his famous tau function r(n), discussed in Chapter 2 of this book, and the Eisenstein series, P, Q, and Ry which are introduced here in Chap- ter 4 and which were key players in so much of his research. In their paper [109], [192, pp. 262-275], Hardy and Ramanujan launched the new field of probabilistic number theory, which became an important branch in number theory. In another paper [110], [192, pp. 276- 309], in the course of obtaining an asymptotic series for the partition function p(n), Hardy and Ramanujan introduced the circle method, which still today is the primary tool for analytically attacking prob- lems in additive number theory. The genesis of the circle method can be found in Ramanujan's notebooks [193], but unfortunately it is most frequently called the Hardy-Littlewood circle method today. In the latter part of his stay in England, Ramanujan wrote his famous papers on congruences for p(n) [188], [192, pp. 210-213] and [190], [192, p. 230], about which much is written in this book. Although Ramanujan never doubted his decision to accept Hardy's invitation to Cambridge, not all was well with Ramanujan. World War I began shortly after his arrival, and being a strict veg- etarian, he could not always obtain familiar food and spices from India. On March 24, 1915, near the end of his first winter in Cam- bridge, Ramanujan wrote his friend E. Vinayaka Row in Madras [51, pp. 116-117], "I was not well till the beginning of this term owing to the weather and consequently I couldn't publish any thing for about 5 months." By the end of his third year in England, Ramanujan was critically ill, and, for the next two years, he was confined to sanitari- ums and nursing homes. Ramanujan's health turned slightly upward when in 1918 he be- came the second Indian to be elected as a Fellow of the Royal Society and the first Indian to be chosen as a Fellow of Trinity College. Af- ter World War I ended, in 1919, Ramanujan returned home, but his health continued to deteriorate, and on April 26, 1920 Ramanujan died at the age of 32. Doctors in both England and India had difficulty diagnosing Ra- manujan's illness. He was treated for tuberculosis, but a severe vi- tamin deficiency, liver cancer, lead poisoning purportedly from not properly cleaning his cooking vessels, and a rare tropical disease were
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