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.