SYNOPSISMath::BigFloat versions`of the Riemann Zeta and Riemann R functions. These are kept in a separate module because they use a lot of big tables that we'd prefer to only load if needed.
DESCRIPTIONPure Perl implementations of Riemann Zeta and Riemann R using Math::BigFloat. These functions are used if:
- The input is a BigInt, a BigFloat, or the bignum module has been loaded.
- The Math::MPFR module is not available.
If you use these functions a lot, I highly recommend you install Math::MPFR, which the main Math::Prime::Util functions will find. These give much better performance, and better accuracy. You can also use Math::Pari for the Riemann Zeta function.
my $z = RiemannZeta($s);
Given a floating point input "s" where "s >= 0.5", returns the floating point value of ζ(s)-1, where ζ(s) is the Riemann zeta function. One is subtracted to ensure maximum precision for large values of "s". The zeta function is the sum from k=1 to infinity of "1 / k^s"
Results are calculated using either Borwein (1991) algorithm 2, or the basic series. Full input accuracy is attempted, but there are defects in Math::BigFloat with high accuracy computations that make this difficult.
my $r = RiemannR($x);
Given a positive non-zero floating point input, returns the floating point value of Riemann's R function. Riemann's R function gives a very close approximation to the prime counting function.
Accuracy should be about 35 digits.
LIMITATIONSBugs in Math::BigFloat (RT 43692, RT 77105) cause many problems with this code. I've attempted to work around them, but it is possible there are cases they miss.
The accuracy goals (35 digits) are sometimes missed by a digit or two.
PERFORMANCEPerformance is quite bad.
AUTHORSDana Jacobsen <[email protected]>
COPYRIGHTCopyright 2012 by Dana Jacobsen <[email protected]>
This program is free software; you can redistribute it and/or modify it under the same terms as Perl itself.