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Efficient implementation of the Hardy-Ramanujan-Rademacher formula. (English) Zbl 1344.11089
Summary: 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. Supplementary materials are available with this article.

MSC:
11Y55 Calculation of integer sequences
11-04 Software, source code, etc. for problems pertaining to number theory
11P81 Elementary theory of partitions
11P83 Partitions; congruences and congruential restrictions
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