Efficient implementation of the Hardy-Ramanujan-Rademacher formula

  title={Efficient implementation of the Hardy-Ramanujan-Rademacher formula},
  author={Fredrik Johansson},
  journal={Lms Journal of Computation and Mathematics},
  • Fredrik Johansson
  • Published 27 May 2012
  • Computer Science
  • Lms Journal of Computation and Mathematics
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… 

On a nonlinear relation for computing the overpartition function

  • M. Merca
  • Mathematics, Computer Science
  • 2020
This formula is combined with a known linear homogeneous recurrence relation for the overpartition function p(n) to obtain a simple and fast computation of the value of p( n).

Estimation of the Partition Number: After Hardy and Ramanujan

The number of conjugate classes of permutations of order n is the same as the partition number p(n). There are already several practical formulae to calculate p(n). But they are either inconvenient

Approximation of the Partition Number After Hardy and Ramanujan: An Application of Data Fitting Method in Combinatorics

Sometimes we need the approximate value of the partition number in a simple and efficient way. There are already several formulae to calculate the partition number p(n). But they are either

Distinct partitions and overpartitions

  • M. Merca
  • Mathematics
    Carpathian Journal of Mathematics
  • 2021
In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function $Q(n)$ which counts partitions of $n$

Generalization of Euler and Ramanujan’s Partition Functions

The theory of partitions has interested some of the best minds since the 18th century. In 1742, Leonhard Euler established the generating function of P(n). Godfrey Harold Hardy said that Srinivasa

Fast and Rigorous Computation of Special Functions to High Precision

This work gives new baby-step, giant-step algorithms for evaluation of linearly recurrent sequences involving an expensive parameter and for computing compositional inverses of power series and shows that isolated values of the integer partition function p(n) can be computed rigorously with softly optimal complexity by means of the Hardy-RamanujanRademacher formula and careful numerical evaluation.

Infinite families of crank functions, Stanton-type conjectures, and unimodality

Dyson’s rank function and the Andrews–Garvan crank function famously give combinatorial witnesses for Ramanujan’s partition function congruences modulo 5, 7, and 11. While these functions can be used

Formulas for the number of partitions related to the Rogers-Ramanujan identities

Abstract In 2011, Santos, Ribeiro and Mondek have obtained a method, using two-line arrays, to representing partitions. Using this method we provide two formulas for the evaluation of the number of

Congruences satisfied by eta-quotients

The values of the partition function, and more generally the Fourier coefficients of many modular forms, are known to satisfy certain congruences. Results given by Ahlgren and Ono for the partition



On the series for the partition function

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 $m$

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

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

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 minimal polynomial of cos(2π/n)

. 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

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

Modular Functions and Dirichlet Series in Number Theory

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

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.