They found an asymptotic series for p(n) -- the number of partitions of n, i.e. the number of ways to write n as a sum of positive integers where you don't count 1+2 and 2+1 as different ways -- with the property that if you take an appropriate number of terms, the nearest integer to the result equals p(n) exactly. Hans Rademacher later tweaked this to give a convergent series that gives p(n) exactly.
The first term of the series (both Hardy&Ramanujan's and Rademacher's) is 1/(4 n sqrt(3)) exp(pi sqrt(2n/3)), which is already a good approximation in the sense that the ratio p(n)/this_approximation(n) tends to 1 as n gets large. This approximation theorem was conjectured by Ramanujan before he came to England, and proved by him and Hardy jointly.
The first term of the series (both Hardy&Ramanujan's and Rademacher's) is 1/(4 n sqrt(3)) exp(pi sqrt(2n/3)), which is already a good approximation in the sense that the ratio p(n)/this_approximation(n) tends to 1 as n gets large. This approximation theorem was conjectured by Ramanujan before he came to England, and proved by him and Hardy jointly.
You can find the actual formulae at http://en.wikipedia.org/wiki/Partition_%28number_theory%29#A... and more details at http://books.google.co.uk/books?id=Sp7z9sK7RNkC&pg=PA68&... .