Computation of the partition function

partitions_rademacher_bound b n N

Sets \(b\) to an upper bound for

\[M(n,N) = \frac{44 \pi^2}{225 \sqrt 3} N^{-1/2} + \frac{\pi \sqrt{2}}{75} \left( \frac{N}{n-1} \right)^{1/2} \sinh\left(\frac{\pi}{N} \sqrt{\frac{2n}{3}}\right).\]

This formula gives an upper bound for the truncation error in the Hardy-Ramanujan-Rademacher formula when the series is taken up to the term \(t(n,N)\) inclusive.

partitions_hrr_sum_arb x n N0 N use_doubles

Evaluates the partial sum \(\sum_{k=N_0}^N t(n,k)\) of the Hardy-Ramanujan-Rademacher series.

If use_doubles is nonzero, doubles and the system's standard library math functions are used to evaluate the smallest terms. This significantly speeds up evaluation for small \(n\) (e.g. \(n < 10^6\)), and gives a small speed improvement for larger \(n\), but the result is not guaranteed to be correct. In practice, the error is estimated very conservatively, and unless the system's standard library is broken, use of doubles can be considered safe. Setting use_doubles to zero gives a fully guaranteed bound.

partitions_fmpz_fmpz p n use_doubles

Computes the partition function \(p(n)\) using the Hardy-Ramanujan-Rademacher formula. This function computes a numerical ball containing \(p(n)\) and verifies that the ball contains a unique integer.

If n is sufficiently large and a number of threads greater than 1 has been selected with flint_set_num_threads(), the computation time will be reduced by using two threads.

See partitions_hrr_sum_arb for an explanation of the use_doubles option.

partitions_fmpz_ui p n

Computes the partition function \(p(n)\) using the Hardy-Ramanujan-Rademacher formula. This function computes a numerical ball containing \(p(n)\) and verifies that the ball contains a unique integer.

partitions_leading_fmpz res n prec

Sets res to the leading term in the Hardy-Ramanujan series for \(p(n)\) (without Rademacher's correction of this term, which is vanishingly small when \(n\) is large), that is, \(\sqrt{12} (1-1/t) e^t / (24n-1)\) where \(t = \pi \sqrt{24n-1} / 6\).