From: "Brian Gladman" <brian.gladman@btinternet.com>
Subject: Re: Zeta computing
Date: Thu, 23 Dec 1999 20:24:26 -0000
Newsgroups: sci.math
Keywords: computing values of the Riemann Zeta function

Max Alekseyev <NO$PAM.relf@i.am> wrote in message
news:945691336@f128.n5015.z2.fidonet.org...
> Hi, All !
>
> What is the fastest way to compute values of the Riemann Zeta function?
> For example, how does Maple V evaluate it?
>
> Bye,
>     Max

Here is a program that can compute the Riemann Zeta Function efficiently
to high accuracy.  It is in ubasic, a multiple precision basic which can be
downloaded from rkmath.rikkyo.ac.jp.   The sample test output is:

   2.0000        1.6449340668482264364724151666460251892190
   3.0000        1.2020569031595942853997381615114499907651
   4.0000        1.0823232337111381915160036965411679027749
   5.0000        1.0369277551433699263313654864570341680573
   6.0000        1.0173430619844491397145179297909205279018
   7.0000        1.0083492773819228268397975498497967595993
   8.0000        1.0040773561979443393786852385086524652574
   9.0000        1.0020083928260822144178527692324120604826
  10.0000       1.0009945751278180853371459589003190170015
  11.0000       1.0004941886041194645587022825264699364626
  12.0000       1.0002460865533080482986379980477396709530
  13.0000       1.0001227133475784891467518365263573957058
  14.0000       1.0000612481350587048292585451051353337383
  15.0000       1.0000305882363070204935517285106450625779
  16.0000       1.0000152822594086518717325714876367220132
  17.0000       1.0000076371976378997622736002935630292028
  18.0000       1.0000038172932649998398564616446219397200
  19.0000       1.0000019082127165539389256569577951013428
  20.0000       1.0000009539620338727961131520386834493354

   Enjoy!

       Brian Gladman

---------------------------------------------------------------
  100   'Riemann Zeta Function Test
  110   point 10:N%=50
  120   dim Zc(N%)
  130   gosub *Zetac
  140   for S=2 to 20
  150     print using(4,4),S,using(6,40),fnZeta(S)
  160   next S
  170   end
  180   '
  190   '
  200   'Calculates values of  the Riemann Zeta Function
  210   'Implemented by Dr B R Gladman, 16 November 1998
  220   '(brian.gladman@btinternet.com).
  230   '
  240   '*zetac must be called first to initialise N%
  250   'values in array Zc() of length N%.     N% is
  260   'about 1.3 times the number of decimal digits
  270   'needed.  The function fnzeta(x) then returns
  280   'the value of the Riemann zeta function using
  290   'the algorithm developed by P. Borwein as set
  300   'out in the paper "An Efficient Algorithm for
  310   'the Riemann Zeta Function". Note that N% and
  320   'Zc() are global variables.
  330   '
  340   *Zetac
  350     local I%,S,T,U
  360     S=1:T=1:U=1
  370     for I%=0 to N%
  380       Zc(I%)=S
  390       T*=(N%-I%)*(N%+I%)*4
  400       U*=(I%+I%+1)*(I%+I%+2)
  410       S+=T/U
  420     next I%
  430     return
  440   fnZeta(X)
  450     local S,I%
  460     Sm=0
  470     for I%=N%-1 to 0 step -1
  480       S=(Zc(N%)-Zc(I%))/(I%+1)^X-S
  490     next I%
  500     return(S/Zc(N%)/(1-2/2^X))
  510   end