A Heuristic Principle and Applications to Gauss Sums

Abstract

In the letter which Ramanujan wrote to Hardy from Madras he claimed, amongst other theorems, an extremely accurate formula for the number of partitions of a number into sums of other natural numbers. In the form which Ramanujan stated it the formula is incorrect but later, in Cambridge, he returned to this question in collaboration with Hardy. They developed a very ingenious method and used it to prove a formula for the partition function of the type that Ramanujan had envisaged. They also used this method to study the Fourier coefficients of various modular forms. Indeed one can now 'explain' the success of this method in the context of modular forms through Petersson's theory of Poincare series.