kevin lucas on Sun, 17 Feb 2019 05:17:30 +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: Sun, 17 Feb 2019 07:17:16 +0300
• Delivery-date: Sun, 17 Feb 2019 05:17:30 +0100
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Thank you. That is extremely interesting. I cannot find a reference or explanation equating Mellin transforms of theta functions and these sums. How did you find this connection?

Kevin

On Sun, Feb 17, 2019 at 5:38 AM Bill Allombert <Bill.Allombert@math.u-bordeaux.fr> wrote:
On Sat, Feb 16, 2019 at 06:04:57PM +0100, Bill Allombert 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 ?

Sorry I meant: the Mellin transform of a theta function,
i.e. something that can be computed with lfunqf.

for example for 2 variables:

? sumnum(a=1,sumnum(b=1,(a^2+b^2)^-4,S))*4 + sumnum(a=1,(a^2)^-4,S)*4
%25 = 4.2814306608057805856207768654374415990
? L=lfunqf(matid(2));
? lfun(L,4)
%27 = 4.2814306608057805856207768654374415990

Cheers,
Bill.