gives a value correct to 14 decimal places. See also the more general Ramanujan–Sato series. One of Ramanujan's remarkable capabilities was the rapid solution of problems, illustrated by the following anecdote about an incident in which P. C. Mahalanobis posed a problem: His intuition also led him to derive some previously unknown identities, such as for all such that and , where is the gamma function, and related to a special value of the Dedekind eta function. Expanding into series of powers and equating coefficients of , , and gives some deep identities for the hyperbolic secant. In 1918, Hardy and Ramanujan studied the partition function extensively. They gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. In 1937, Hans Rademacher refined their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae called the circle method. In the last year of his life, Ramanujan discovered mock theta functions. For many years, these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms. The Ramanujan conjecture Although there are numerous statements that could have borne the name Ramanujan conjecture, one was highly influential on later work. In particular, the connection of this conjecture with conjectures of André Weil in algebraic geometry opened up new areas of research. That Ramanujan conjecture is an assertion on the size of the tau-function, which has as generating function the discriminant modular form Δ(q), a typical cusp form in the theory of modular forms. It was finally proven in 1973, as a consequence of Pierre Deligne's proof of the Weil conjectures. The reduction step involved is complicated. Deligne won a Fields Medal in 1978 for that work. In his paper "On certain arithmetical functions", Ramanujan defined the so-called delta-function, whose coefficients are called (the Ramanujan tau function). He proved many congruences for these numbers, such as for primes . This congruence (and others like it that Ramanujan proved) inspired Jean-Pierre Serre (1954 Fields Medalist) to conjecture that there is a theory of Galois representations that "explains" these congruences and more generally all modular forms. is the first example of a modular form to be studied in this way. Deligne (in his Fields Medal-winning work) proved Serre's conjecture. The proof