Re: Calculating zeta(1+it) for t in range of 10^12 accurately

*To*: mathgroup at smc.vnet.net*Subject*: [mg106813] Re: Calculating zeta(1+it) for t in range of 10^12 accurately*From*: "Steve Luttrell" <steve at _removemefirst_luttrell.org.uk>*Date*: Sun, 24 Jan 2010 05:40:07 -0500 (EST)*References*: <hj130p$bd4$1@smc.vnet.net>

On http://reference.wolfram.com/mathematica/note/SomeNotesOnInternalImplementation.html#21479 it says: "Zeta and related functions use Euler-Maclaurin summation and functional equations. Near the critical strip they also use the Riemann-Siegel formula." -- Stephen Luttrell West Malvern, UK "Dominic" <miliotodc at rtconline.com> wrote in message news:hj130p$bd4$1 at smc.vnet.net... > Hello guys. I am working on a project which requires me to calculate > zeta(1+it) for t in the range no more than 10^12 and it's in > arbitrary-precision format like: > > t=(Pi 10^12)/Log(3) -1/3; > > What is the best way to make sure I'm getting the accuracy correctly to > at least 6 decimal places? I've been using: > > N[Zeta[1+it],{Infty,20}] > > Is this the best way? Also, can anyone tell me what algorithm > Mathematica is using to calculate zeta at this value? Is it > Riemann-Segel? > > Thanks, > > Dominic > >