       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
>
>

```

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