Fourier Coefficients of Harmonic Weak Maass Forms and the Partition Function

In a recent paper, Bruinier and Ono proved that certain harmonic weak Maass forms have the property that the Fourier coefficients of their holomorphic parts are algebraic traces of weak Maass forms evaluated on Heegner points. As a special case they obtained a remarkable finite algebraic formula for the Hardy-Ramanujan partition function p(n), which counts the number of partitions of a positive integer n. We establish an asymptotic formula with a power saving error term for the Fourier coefficients in the Bruinier-Ono formula. As a consequence, we obtain a new asymptotic formula for p(n). One interesting feature of this formula is that the main term contains essentially 3 · h(−24n + 1) fewer terms than the truncated main term in Rademacher’s exact formula for p(n), where h(−24n+ 1) is the class number of the imaginary quadratic field Q( √ −24n + 1).