The Partition Function Revisited

In 1918, Hardy and Ramanujan wrote their landmark paper deriving the asymptotic formula for the partition function. The paper however was fundamental for another reason, namely for introducing the circle method in questions of additive number theory. Though this method is powerful, it is often difficult and technically complicated to employ. In 2011, Bruinier and Ono discovered a new algebraic formula for the partition function obtained via the theory of weak Maass forms. This formula allows us to deduce the Hardy-Ramanujan formula using basic theory of Fourier expansions of Maass forms and the theory of positive definite binary quadratic forms. The Hardy-Ramanujan formula also leads to the asymptotics of Fourier coefficients of the j -function, a fact hitherto unnoticed. These asymptotics were obtained earlier by Petersson in 1932 and Rademacher in 1938 (independently) using the circle method. We report on our joint work with M. Dewar in this context.