Re: Taylor series of the zeta function
- To: mathgroup at smc.vnet.net
- Subject: [mg103328] Re: Taylor series of the zeta function
- From: Szabolcs Horvát <szhorvat at gmail.com>
- Date: Tue, 15 Sep 2009 06:54:08 -0400 (EDT)
- References: <h8nj02$cde$1@smc.vnet.net>
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/SomeNotesOnInternalImplementation.html