acb_mzeta_mzv

Sets res to the multiple zeta value

with the convention of PARI/GP's zetamult (the first index belongs to the largest summation variable; $s_1 \ge 2$ for convergence, otherwise the result is indeterminate). The empty index gives 1, depth 1 uses arb_zeta_ui.

For depth $r \ge 2$ the value is computed from the iterated integral representation by Akhilesh's double tails [Akh2017], a recursion over the subwords of the index in fixed-point arithmetic, with cost $O(k^2 P^2)$ for weight $k$ and $P$ bits; from 25000 bits it is instead reduced by Hölder convolution [BBBL2001] to multiple polylogarithms at $\tfrac12$, evaluated by binary splitting in quasi-linear time. See Double tails and Hölder convolution.