NAME

       Math::Prime::Util::ZetaBigFloat - Perl Big Float versions of Riemann Zeta and R functions

VERSION

       Version 0.73

SYNOPSIS

       Math::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.

DESCRIPTION

       Pure 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::Prime::Util::GMP module is not available or old.

       If you use these functions a lot, I highly recommend you install Math::Prime::Util::GMP,
       which the main Math::Prime::Util functions will find.  These give much better performance,
       and better accuracy.  You can also use Math::Pari and Math::MPFR for the Riemann Zeta
       function.

FUNCTIONS

   RiemannZeta
         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.

   RiemannR
         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.

LIMITATIONS

       Bugs 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.

PERFORMANCE

       Performance is quite bad.

SEE ALSO

       Math::Prime::Util

       Math::Prime::Util::GMP

       Math::MPFR

       Math::Pari

AUTHORS

       Dana Jacobsen <dana@acm.org>

COPYRIGHT

       Copyright 2012 by Dana Jacobsen <dana@acm.org>

       This program is free software; you can redistribute it and/or modify it under the same
       terms as Perl itself.