kevin lucas on Sat, 16 Feb 2019 20:52:38 +0100

 Re: Evaluating Multiple Sums in PARI/GP

• To: pari-users@pari.math.u-bordeaux.fr
• Subject: Re: Evaluating Multiple Sums in PARI/GP
• Date: Sat, 16 Feb 2019 22:52:23 +0300
• Delivery-date: Sat, 16 Feb 2019 20:52:38 +0100
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I don't know of a way to write this sum using theta functions. There is a more elementary result due to Hardy, that
sum(a^2+b^2)^-s = zeta(s)*dirichletbeta(s) - zeta(2s)
so I suspect the more general sums being considered would require either multiple zetas or some clever resummation based on this.

Kevin

On Sat, Feb 16, 2019 at 8:05 PM Bill Allombert <Bill.Allombert@math.u-bordeaux.fr> wrote:
On Fri, Feb 15, 2019 at 08:28:37PM +0300, kevin lucas wrote:
> I recently ran into problems attempting to formulate a PARI program that
> evaluated the _expression_
>
> sum(((-1)^(a+b+c))/(a^2 + b^2 + c^2)^s)
>
> for various complex values of s, with a,b,c running over Z^3/{(0,0,0)}. How
> should I attempt this? More generally, how should one set up iterated
> alternating sums like these? If, for instance I also wanted the
> eight-dimensional version of the above sum, how would I compute it?

Is it not some kind of theta function ?
Then you could use lfunqf (with Q=matid(8)).

Cheers,
Bill