Rademacher's Series for the Partition Function

This entry provides a formalisation of Rademacher's convergent series for the partition function. The partition function $p(n)$ is defined as the number of ways in which a natural number $n$ can be written as the sum of positive integers, without taking the order into account. E.g. $p(3) = 3$ since $3 = 2 + 1 = 1 + 1 + 1$.

Using a contour integration argument based on earlier asymptotic sums for $p(n)$, Rademacher found the remarkable convergent series representation \[p(n) = \frac{1}{\pi\sqrt{2}} \sum_{k\geq 1} A_k(n) \sqrt{k} f(n,k)\] where \[A_k(n) = \hskip-1em\sum_{\substack{1\leq h\leq k\\\text{gcd}(h,k)=1}}\hskip-1em \cos(\pi(s(h,k) - 2nh/k))\] \[f(x,k) = \frac{\text{d}}{\text{d}x}\, \frac{\sinh\left[\tfrac{\pi}{k}\sqrt{\tfrac{2}{3}(x-\tfrac{1}{24})}\,\right]} {\sqrt{x-\tfrac{1}{24}}}\] and $s(h,k)$ denotes a Dedekind sum. Decent (albeit not very tight) bounds for the remainder of the truncated sum are also derived.

One consequence of this formula that was also formalised is the asymptotic estimate $p(n) \sim \exp(\pi\sqrt{2/3\,n}) / (4\sqrt{3}\,n)$.

Rademacher's series also forms the basis for the most efficient known algorithms to compute $p(n)$, but this requires significantly more work and is out of scope for this entry.