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Re: Taylor series of the zeta function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg103346] Re: Taylor series of the zeta function
  • From: pfalloon <pfalloon at gmail.com>
  • Date: Wed, 16 Sep 2009 05:48:22 -0400 (EDT)
  • References: <h8nj02$cde$1@smc.vnet.net> <h8nrm3$3kc$1@smc.vnet.net>

On Sep 15, 8:52 pm, Szabolcs Horv=E1t <szhor... at gmail.com> wrote:
> On 2009.09.15. 10:24, Guy Verhofstadt wrote:
>
> > Hi
> > I would like to compute the Taylor series of the (logarithm) of the
> > Riemann zeta function at various integral points, up to high order and
> > with high precision.
> > Mathematica does quite well at this:
> > N[Series[Log[Zeta[x]], {x, 85, 30}], 100]
> > gives a result, for instance.
> > I would like to know which algorithm is used to compute this, or how I
> > could find out.
> > Thank you
>
> There is a page in the documentation which has a few notes on the
> implementation of various functions (but not much).  If Series works in
> the obvious way, the challenge is computing the Zeta function and its
> derivative.  About Zeta the page says:
>
> "Zeta and related functions use Euler-Maclaurin summation and functional
> equations. Near the critical strip they also use the Riemann-Siegel
> formula."
>
> http://reference.wolfram.com/mathematica/note/SomeNotesOnInternalImpl...

The Wolfram special functions website gives useful formulas that may
help with what you're trying to do. In particular, there is a formula
for the nth derivative of Zeta[x]:

http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/20/02/01/

Cheers,
Peter.


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