2012/05/31 by Fredrik Johansson · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics #Congruence relation #Discrete mathematics #Exponent #Mathematics #Partition (number theory) #Ramanujan's sum #math.NT #msc:11P83 #msc:11Y55
paper · pdf · doi:10.1112/s1461157012001088
published as LMS J. Comput. Math. 15 (2012) 341-359 · updated version containing an unconditional complexity proof; accepted for publication in LMS Journal of Computation and Mathematics
arxiv created 2012/07/06 · openalex publication_date 2012/10/01 · arxiv updated 2019/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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(n1/2+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.