Equidistribution of Heegner points and the partition function

Let p(n) denote the number of partitions of a positive integer n. In this paper we study the asymptotic growth of p(n) using the equidistribution of Galois orbits of Heegner points on the modular curve X ₀(6). We obtain a new asymptotic formula for p(n) with an effective error term which is O(n^-(\frac12+d)){O(n^{-(\frac{1}{2}+\delta)})} for some δ > 0. We then use this asymptotic formula to sharpen the classical bounds of Hardy and Ramanujan, Rademacher, and Lehmer on the error term in Rademacher’s exact formula for p(n).