From: "Dik T. Winter" <Dik.Winter@cwi.nl>
Subject: Re: Computing Riemann Zeta function
Date: Mon, 6 Mar 2000 15:57:24 GMT
Newsgroups: sci.math
Summary: [missing]

In article <38C13B01.E1CB22D8@club-internet.fr> Raymond Manzoni <raymman@club-internet.fr> writes:
 > A nice historical introduction may be found in 
 > "Computational Number Theory at CWI in 1970-1994"
 > but the paper (Riele-Lune.ps) doesn't seem available online any longer.

It is available again (in compressed form):
  http://www.cwi.nl/ftp/herman/papers/CNTatCWI70-94.ps.Z
-- 
dik t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~dik/
==============================================================================

From: munafo@gcctech.com
Subject: Re: Computing Riemann Zeta function
Date: Tue, 07 Mar 2000 05:33:19 GMT
Newsgroups: sci.math

Yes, Raymond -- this is just what I was looking for -- thank you!

It will take a few days to digest but it looks like I'll start with PARI
and then look at the Euler MacLaurin series you outlined.

Thanks!  - Robert

  Raymond Manzoni <raymman@club-internet.fr> wrote:
>[various references including:]
> At CECM you'll find :
> "Computational strategies for the Riemann zeta function" at
>     http://www.cecm.sfu.ca/preprints/1998pp.html#98:118
> consult too Andrew Odlyzko's "Papers on Zeros of the Riemann Zeta
> Function and Related Topics" at
> http://www.research.att.com/~amo/doc/zeta.html
> (for example "Fast algorithms for multiple evaluations of the Riemann
> zeta function")
> [...]
> Or forget all that and use simply the quick pari/gp for evaluation of
> zeta in the critical strip (available with C source at
> http://www.parigp-home.de/ )
> [...] Euler Mac-Laurin [gives]:
>
>     zeta(x) ~ sum(1/k^x, k=1..N)
>              + 1/((x-1)*N^(x-1))
>              -1/(2*N^x)
>              +x/(12*N^(x+1))
>              -x*(x+1)*(x+2)/(720*N^(x+3))
>              +x*(x+1)*(x+2)*(x+3)*(x+4)/(30240*N^(x+5))
>              +...
>
>   The greater the N the better the result (but even N=10 is good)
>   The accuracy is rather fine when  |Im(x)| < 2*PI*N and
>   becomes very bad after that! [...]


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