Efficient implementation of the Hardy-Ramanujan-Rademacher formula

TL;DR

This paper presents an efficient implementation of the Hardy-Ramanujan-Rademacher formula for computing the partition function p(n), achieving significant speedups and enabling calculations for very large n, with applications in discovering new congruences.

Contribution

The paper introduces a new implementation that computes p(n) with near-optimal complexity, vastly improves speed over previous software, and extends the known congruences for the partition function.

Findings

Achieved over 500x speedup in computing p(n)

Successfully computed p(10^{19}) for the first time

Discovered over 22 billion new congruences for p(n)

Abstract

We describe how the Hardy-Ramanujan-Rademacher formula can be implemented to allow the partition function $p(n)$ to be computed with softly optimal complexity $O(n^{1/2+o(1)})$ and very little overhead. A new implementation based on these techniques achieves speedups in excess of a factor 500 over previously published software and has been used by the author to calculate $p(10^{19})$, an exponent twice as large as in previously reported computations. We also investigate performance for multi-evaluation of $p(n)$, where our implementation of the Hardy-Ramanujan-Rademacher formula becomes superior to power series methods on far denser sets of indices than previous implementations. As an application, we determine over 22 billion new congruences for the partition function, extending Weaver's tabulation of 76,065 congruences.