A Rademacher Type Formula for Partitions and Overpartitions

A Rademacher-type convergent series formula which generalizes the Hardy-Ramanujan-Rademacher formula for the number of partitions of n and the Zuckerman formula for the Fourier coefficients of θ4(0 | τ)−1 is presented. 1. Background 1.1. Partitions. A partition of an integer n is a representation of n as a sum of positive integers, where the order of the summands (called parts) is considered irrelevant. It is customary to write the parts in nonincreasing order. For example, there are three partitions of the integer 3, namely 3, 2 + 1, and 1 + 1 + 1. Let p(n) denote the number of partitions of n, with the convention that p(0) = 1, and let f(x) denote the generating function of p(n), i.e. let