VI.82 Srinivasa Ramanujan

b. Erode, India, 1887; d. Madras (now Chennai), India, 1920 Partitions; modular forms; mock theta functions

Ramanujan, a self-taught Indian genius, made monumental contributions to mathematics that set the stage for many of the breakthroughs in number theory in the twentieth century. He worked on analytic number theory, as well as on ELLIPTIC FUNCTIONS [V.31], hypergeometric series, and the theory of CONTINUED FRACTIONS [III.22]. Much of this work was carried out together with his friend, benefactor, and collaborator G. H. HARDY [VI.73].

Hardy and Ramanujan founded the powerful “circle method” in their remarkable paper that gave an exact formula for p(n), the number of integer partitions of n. Ramanujan independently discovered the two identities that came to be known as the Rogers–Ramanujan identities:

These have applications ranging from LIE THEORY [III.48] to statistical physics. The importance of these identities relates to the fact that the GENERATING FUNCTION [IV.18 §§2.4, 3] for p(n) is

Thus, for example, the second identity asserts that the number of partitions of n into parts all of which are 2 or 3 mod 5 is equal to the number of partitions into distinct parts, all greater than 1, in which no two parts are consecutive integers.

In his work on p(n) Ramanujan discovered and proved many divisibility properties, e.g., that 5 always divides p(5n + 4) and that 7 always divides p(7n + 6). His conjectures on these divisibility properties inspired the development of extensive methods in MODULAR FORMS [III.59], and his last conjecture was finally settled in 1969 by Oliver Atkin.

All Ramanujan’s studies involving p(n) concerned the modular form

The relevance of this is that q^1/124/η (w) is the generating function for p (n). Of special interest to Ramanujan was the arithmetic function τ (n), defined by the 24th power of η (w): namely,

Ramanujan conjectured that | τ (p) | < 2ρ^11/2 for every prime p. The study of this problem led to deep and extensive work on modular forms by H. Petersson, R. Rankin, and others. Eventually, the conjecture was proved by P. Deligne, who received the Fields Medal for his achievement in 1978.

The full story of Ramanujan’s life makes his achievements all the more amazing. As a child he was mathematically precocious. In high school he won prizes in mathematics. On the basis of his high school record, he won a scholarship to the Government College in Kumbakonam in 1904. At about this time, Ramanujan came into contact with the book A Synopsis of Elementary Results in Pure and Applied Mathematics by G. S. Carr. This rather eccentric book is essentially a huge collection of formulas and theorems compiled for students preparing for the celebrated Mathematical Tripos examination at Cambridge. This book fascinated Ramanujan, who became obsessed with mathematics. In college, he neglected his other subjects and gave his all to mathematics. Consequently, he failed some subjects and lost his scholarship. By 1913, Ramanujan seemed destined for obscurity—he was now a mere clerk in the Madras Port Trust. Friends encouraged him to write to English mathematicians about his mathematical discoveries. Eventually he wrote to G. H. Hardy, who was able to discern that Ramanujan was a truly extraordinary mathematician.

Hardy arranged for Ramanujan to travel to England, and between 1914 and 1918 the two of them produced the groundbreaking work described above.

In 1918, Ramanujan became ill with a sickness diagnosed as tuberculosis. He convalesced in England for a year. His health improved a little in 1919 and he was able to return to India. Unfortunately, his health worsened after his return, and he died in 1920. During this last year in India he penned the pages now known as Ramanujan’s Lost Notebook and therein laid the foundations of the theory of mock theta functions, a class of functions similar to but more general than the classical theta functions.

Further Reading

Berndt, B. 1985–98. Ramanujan’s Notebooks. New York: Springer.

Kanigel, R. 1991. The Man Who Knew Infinity. New York: Scribners.

George Andrews