idea 1 for Besançon atelier

[Vincent Delecroix <vincent.delecroix@u-bordeaux.fr>]

Dear pari devs,

Here is one thing I would like to investigate and in which PARI/GP (or 
their developers) might be of some help.

I got interested in some generalizations of multiple zeta values (and 
relations between them). My sums look like

 sum_{h1, h2, ..., hm}  1 / L1(h) L2(h) ... Lm(h)

where
 * the sum is over all positive integers h1, ..., hm
 * Li(h) are linear with coefficients 0 or 1

Example of such things are the multiple zeta values as

zeta(s1, ..., sm) =
    sum 1 / (h1^s1 (h1 + h2)^s2 ... (h1 + ... + hm)^sm)

I tried with more or less success to get numerical approximations in 
PARI/GP (sumnummonien is sometimes very wrong and sumnum is sometimes 
very slow). Due to the very specific form of my numbers I believe that 
there are more clever algorithms.

In case I succeded to get approximations, I just use the very nice 
lindep function to check relations with standard multizetas or product 
of zeta.

In some very elementary cases, it is even possible to split the sum 
(looking at linear constraints in the hi) in order to make a sum of 
multiple zeta like

sum 1 / h1^s1 h2^s2 = zeta(s1,s2) + zeta(s2,s1) + zeta(s1+s2)

Best
Vincent