Srinivasa Ramanujan,
who passed away on 26 April 1920, aged 32, in Kumbakonam, was an Indian
mathematician whose contributions to the theory of numbers include
pioneering discoveries of the properties of the partition function.
When he was 15 years old, he
obtained a copy of George Shoobridge Carr’s Synopsis of
Elementary Results in Pure and Applied Mathematics, 2 vol.
(1880–86). This collection of thousands of theorems, many presented with
only the briefest of proofs and with no material newer than 1860, aroused his
genius. Having verified the results in Carr’s book, Ramanujan went beyond it,
developing his own theorems and ideas. In 1903 he secured a scholarship to the University
of Madras but lost it the following year because he neglected all other studies
in pursuit of mathematics.
Ramanujan continued his work,
without employment and living in the poorest circumstances. After marrying in
1909 he began a search for permanent employment that culminated in an interview
with a government official, Ramachandra Rao. Impressed by Ramanujan’s
mathematical prowess, Rao supported his research for a time, but Ramanujan,
unwilling to exist on charity, obtained a clerical post with the Madras Port
Trust.
In 1911 Ramanujan published the first of his papers in the Journal of the Indian Mathematical Society. His genius
slowly gained recognition, and in 1913 he began a correspondence with the
British mathematician Godfrey
H. Hardy that led to a special scholarship from the University of Madras
and a grant from Trinity College, Cambridge. Overcoming his religious
objections, Ramanujan traveled to England in 1914, where Hardy tutored him and collaborated
with him in some research.
Ramanujan’s knowledge of
mathematics (most of which he had worked out for himself) was startling.
Although he was almost completely unaware of modern developments in
mathematics, his mastery of continued fractions was unequaled by any
living mathematician. He worked out the Riemann series, the elliptic integrals,
hypergeometric series, the functional equations of the zeta function, and
his own theory of divergent series, in which he found a value for the sum of
such series using a technique he invented that came to be called Ramanujan
summation. On the other hand, he knew nothing of doubly periodic functions, the
classical theory of quadratic forms, or Cauchy’s theorem, and he had only the
most nebulous idea of what constitutes a mathematical proof.
Though brilliant, many of his theorems on the theory of prime numbers were
wrong.
In England Ramanujan made further advances, especially in the
partition of numbers (the number of ways that a positive integer can be
expressed as the sum of positive integers; e.g., 4 can be expressed as 4, 3 +
1, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1). His papers were published in English
and European journals, and in 1918 he was elected to the Royal Society of
London. In 1917 Ramanujan had contracted tuberculosis, but his condition
improved sufficiently for him to return to India in 1919. He died the
following year, generally unknown to the world at large but recognized by
mathematicians as a phenomenal genius, without peer since Leonhard Fuler (1707–83)
and Carl Jacobi (1804–51). Ramanujan left behind three notebooks and a
sheaf of pages (also called the “lost notebook”) containing many unpublished
results that mathematicians continued to verify long after his death.
