# Efficient implementation of the Hardy-Ramanujan-Rademacher formula

@article{Johansson2012EfficientIO, title={Efficient implementation of the Hardy-Ramanujan-Rademacher formula}, author={Fredrik Johansson}, journal={Lms Journal of Computation and Mathematics}, year={2012}, volume={15}, pages={341-359} }

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…

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## References

SHOWING 1-10 OF 47 REFERENCES

### On the series for the partition function

- Mathematics
- 1938

Hardy and Ramanujan were unable to decide several questions about (1.1). For instance, if a is given, (1.1) gives /»(ra) to within half a unit for all sufficiently large ra. Just how large ra must be…

### Distribution of the partition function modulo $m$

- Mathematics
- 2000

Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin,…

### On a conjecture of Ramanujan

- Mathematics
- 1936

BY ~lsing the Hardy-Ramanujan asymptotic formula for p(n) the number of unrestricted partitions of n, true for large values of n, Lehmer 1 has recently found that (1) Ÿ ----4353 50207 84031 73482…

### New Congruences for the Partition Function

- Mathematics
- 2001

AbstractLet p(n) denote the number of unrestricted partitions of a non-negative integer n. In 1919, Ramanujan proved that for every non-negative n
$$\begin{gathered} p(5 + 4) \equiv 0(\bmod 5),…

### The average least quadratic nonresidue modulo m and other variations on a theme of Erdős

- Mathematics
- 2012

### The minimal polynomial of cos(2π/n)

- Mathematics
- 1993

. It is well known that if Φ n ( x ) is the n th cyclotomic polynomial, then there is a factorization x n − 1 = Q Φ d ( x ), where the product is taken over the divisors d of n . Thus, one can…

### A root of unity occurring in partition theory

- Mathematics
- 1970

In this paper a new representation is found for the root of unity occurring in the well-known transformation equation of the generating function for p(n), the number of partitions of the positive…

### Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms

- Mathematics
- 2011

### Modular Functions and Dirichlet Series in Number Theory

- Mathematics
- 1976

This is the second volume of a 2-volume textbook which evolved from a course (Mathematics 160) offered at the California Institute of Technology du ring the last 25 years. The second volume…

### Computing the integer partition function

- Computer Science, MathematicsMath. Comput.
- 2007

Efficient algorithms for computing the values of the partition function are discussed and implemented and these algorithms are implemented in order to conduct a numerical study of some conjec- tures related to the partitions function.