Vincent Delecroix on Fri, 06 Oct 2017 22:00:47 +0200

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idea 1 for Besançon atelier

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)

 * 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)