Dr. Robert Harley on Tue, 6 May 2003 16:31:42 +0200 (CEST) |
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Re: Rademacher formula for p(n) with PARI |
Hi all, Bill wrote: > [...] > * This program is basically the implementation of the script . * [...] > * part(n) = round(sum(q=1,max(5,0.24*sqrt(n)+2),L(n,q)*Psi(n,q))) Hmm... that formula doesn't take enough terms of Rademacher's series, when n is small. In particular it gives p(244) = 144867692496446 instead of 144867692496445 and p(259) = 459545750448674 instead of 459545750448675. Checking in the source of 2.2.5, what it actually uses is: max = (ulong)(sqrt(gtodouble(n) ) * 0.24 + 5); which is enough. For large n (more than about 250K) the number of terms can be reduced using Thm 14 from Lehmer's paper (as in the script I posted). This can make a noticeable difference since the time to compute the terms grows quickly. Perhaps this could be incorporated into Ralf's code to speed it up for such arguments? (No, I'm not volunteering to do so! :) Regards, Rob. .-. .-. / \ .-. .-. / \ / \ / \ .-. _ .-. / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / `-' `-' \ / \ / \ \ / `-' `-' \ / `-' `-'