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

 idea 1 for Besançon atelier

```Dear pari devs,

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

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