Function: zetamult
Section: transcendental
C-Name: zetamult_interpolate
Prototype: GDGp
Help: zetamult(s,{t=0}): multiple zeta value at integral s = [s1,...,sk];
 more generally, return Yamamoto's t-MZV interpolation (star value for t = 1).
Doc: For $s$ a vector of positive integers such that $s[1] \geq 2$,
 returns the multiple zeta value (MZV)
 $$\zeta(s_{1},\dots, s_{k}) = \sum_{n_{1}>\dots>n_{k}>0}
    n_{1}^{-s_{1}}\dots n_{k}^{-s_{k}}$$
 of length $k$ and weight $\sum_{i} s_{i}$.
 More generally, return Yamamoto's $t$-MZV interpolation evaluated at $t$:
 for $t = 0$, this is the ordinary MZV; for $t = 1$, we obtain the MZSV
 star value, with $\geq$ instead of strict inequalities;
 and of course, for $t = \kbd{'x}$ we obtain Yamamoto's one-variable polynomial.
 \bprog
 ? zetamult([2,1]) - zeta(3) \\ Euler's identity
 %1 = 0.E-38
 ? zetamult([2,1], 1)   \\ star value
 %2 = 2.4041138063191885707994763230228999815
 ? zetamult([2,1], 'x)
 %3 = 1.20205[...]*x + 1.20205[...]
 @eprog\noindent
 If the bit precision is $B$, this function runs in time
 $\tilde{O}(k(B+k)^{2})$ if $t = 0$, and $\tilde{O}(kB^{3})$ otherwise.

 In addition to the above format (\kbd{avec}), the function
 also accepts a binary word format \kbd{evec} (each $s_{i}$ is replaced
 by $s_{i}$ bits, all of them 0 but the last one) giving the MZV
 representation as an iterated integral, and an \kbd{index} format
 (if $e$ is the positive integer attached the \kbd{evec} vector of
 bits, the index is the integer $e + 2^{k-2}$). The function
 \kbd{zetamultconvert} allows to pass from one format to the other; the
 function \kbd{zetamultall} computes simultaneously all MZVs of weight
 $\sum_{i\leq k} s_{i}$ up to $n$.
Variant: Also available is \fun{GEN}{zetamult}{GEN s, long prec} for $t = 0$.
