Zeta-like multizeta values for $\mathbb{F}_q [t]$

Multizeta values for function fields: A survey

“…Conjecturally these are all the primitive eulerian tuples in depth more than 2. The last conjecture was made in this simple explicit form in [15] by checking on the extensive data generated by the efficient algorithm constructed there, but is immediately seen to be equivalent (as mentioned in [30 3 ) are eulerian by the tuple restriction conjecture 7.1 (2), but the second co-ordinate of the first two in depth 2 list cannot be the first co-ordinate of eulerian (even up to p-powers), so that (s 1 , s 2 ) = (q n − 1, (q − 1)q n ) for some n, without loss of generality. By induction on r > 2, S is T r−1 followed by s r .…”

“…Thus, it was speculated that, apart from the shuffle relations effects, the iterated indices s i , i > 1 in the multizeta linear relations should be "even". This is true e.g., at S d level in the sum shuffle relations above and for the Euler type relations and the eulerian and zeta-like relations below, because the depth reduction mechanism (see e.g., [49, 3.4.6] or proofs in [30]) seems to come through the special cancellations in the products of such functions. See also [13, Thm.…”

“…We discuss the results, conjectures and algorithms to decide when the ratio of a multizeta value with a zeta value is algebraic [12,15,18,25,30]. As we now know, by Chang's results (only expected when the definition was made) recalled below in Section 9, that no such relation exists, in different weights, and that the ratio is algebraic if and only if it is rational, we make the following definitions.…”

“…Definitions. [30,47] We We take this opportunity to mention that in Conjecture 4.4 of [30] which gives full conjectural list of the zeta-like values of depth at most q 2 , the word "primitive" should have been dropped and that the remaining part of 4.4 also follows from Conjecture 7.1 (1) (i.e. Conjecture 4.1 (i) of [30]) and the parity conjecture mentioned in §6.4, which as mentioned there, follows in this case from the result of Chang.…”

“…We had only checked and expected (by the parity conjecture mentioned above) the second statement on our data. In [15] ( [25] respectively), efficient algorithms to effectively check whether a given tuple is eulerian (zeta-like respectively), for a given q , were developed using motivic interpretation ideas and were used to produce much more (than [30]) data, which was then used to verify and make conjectures.…”