Re: Evaluating Multiple Sums in PARI/GP

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Sun, Feb 17, 2019 at 07:17:16AM +0300, kevin lucas wrote:
> 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?

The Mellin transform converts modular forms to L functions.
The basic formula is:
M(exp(-n*s)) = gamma(s)/n^s

So if you set q = exp(-s), then
M(Sum a_n*q^n) = gamma(s)*Sum(a_n/n^s))

If Sum a_n*q^n is a modular form of integral weight, 
then the right hand is a L function that can be complted to a Lambda
function that satisfy the usual functional equation
'Lambda(s)=Lambda(2-s)', and so can be computed with lfun.

So for Sum (a_1^2+...+a_8^2)^-s can be computed with
L=lfunqf(matid(8)); lfun(L,s)

If you add an additive character, I think you can deal with it by using
inflated quadratic forms.

For example for
Sum (-1)^a*(a^2+b^2)^-s

L1=lfunqf([1,0;0,1]); L2=lfunqf([4,0;0,1]);
s1 = 2*lfun(L2,4)-lfun(L1,4)
{
  s2 = sumalt(a=1,(-1)^a*sumnum(b=1,(a^2+b^2)^-4))*4+
         sumalt(a=1,(-1)^a*(a^2)^-4)*2+sumalt(b=1,(b^2)^-4)*2
}
%4 = -0.23414073926281612577613623482861008743
%5 = -0.23414073926586462589242355288509555826

Cheers,
Bill.