Re: Rademacher formula for p(n) with PARI

[Bill Allombert <allomber@math.u-bordeaux.fr>]

On Tue, May 06, 2003 at 07:08:48AM +0200, Damien Wyart wrote:
> Hello,
> 
> I am quite new to PARI (and this list) and plan to implement
> Rademacher's formula for p(n) using PARI. As this is a classical result,
> maybe some people here have already done this. Any hints or code
> snippets would then be very welcome.

PARI 2.2.5 development version include a function numbpart
that implement Hardy-Ramanujan-Rademacher's formula.

? numbpart(17)
%1 = 297

Here the comment which include a GP implementation:

/* Original code contributed by: Ralf Stephan (ralf@ark.in-berlin.de).
 *
 * This program is basically the implementation of the script
 *
 * Psi(n, q) = local(a, b, c); a=sqrt(2/3)*Pi/q; b=n-1/24; c=sqrt(b);
 *             (sqrt(q)/(2*sqrt(2)*b*Pi))*(a*cosh(a*c)-(sinh(a*c)/c))
 * L(n, q) = if(q==1,1,sum(h=1,q-1,if(gcd(h,q)>1,0,cos((g(h,q)-2*h*n)*Pi/q))))
 * g(h, q) = if(q<3,0,sum(k=1,q-1,k*(frac(h*k/q)-1/2)))
 * part(n) = round(sum(q=1,max(5,0.24*sqrt(n)+2),L(n,q)*Psi(n,q)))
 */

Cheers,
Bill.