LMS Journal of Computation and Mathematics

Research Article

Efficient implementation of the Hardy–Ramanujan–Rademacher formula

Fredrik Johanssona1

a1 Research Institute for Symbolic Computation, Johannes Kepler University, Altenberger Strasse 69, 4040 Linz, Austria (email: [email protected])


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(1019), 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. Supplementary materials are available with this article.

(Received October 31 2011)

(Revised June 01 2012)

(Online publication October 2012)

2010 Mathematics Subject Classification

  • 11Y55 (primary);
  • 11-04;
  • 11P83 (secondary)


Supported by Austrian Science Fund (FWF) grant Y464-N18 (Fast Computer Algebra for Special Functions).