FLINT3.7.0-dev

acb_mzeta_polylog

Synopsis#include <flint/acb_mzeta.h>
void acb_mzeta_polylog(acb_t res, const ulong * s, acb_srcptr x, slong r, slong prec)source

Sets res to the multiple polylogarithm

Li⁡s1,…,sr(x1,…,xr)=∑n1>⋯>nr≥1x1n1⋯xrnrn1s1⋯nrsr\operatorname{Li}_{s_1, \ldots, s_r}(x_1, \ldots, x_r) = \sum_{n_1 > \cdots > n_r \ge 1} \frac{x_1^{n_1} \cdots x_r^{n_r}}{n_1^{s_1} \cdots n_r^{s_r}}

(the convention of PARI/GP's polylogmult). The value is computed from Goncharov's iterated integrals G(a1,…,aL;1)G(a_1, \ldots, a_L; 1) with letters ai∈{0,1/y1,…,1/yr}a_i \in \{0, 1/y_1, \ldots, 1/y_r\}, yi=x1⋯xiy_i = x_1 \cdots x_i, as power series at 0 by binary splitting, either directly (when all nonzero letters have absolute value at least 2) or after splitting the path at 12\tfrac12 (when the nonzero letters aia_i and 1−ai1 - a_i have absolute value at least 34\tfrac34); see Multiple polylogarithms. Other arguments, for example xx outside the unit disk or letters close to 12\tfrac12, are not supported yet, and the result is then indeterminate, as it is for divergent sums (s1=1s_1 = 1, x1=1x_1 = 1).

Cite this page

The FLINT team, “acb_mzeta_polylog (acb_mzeta.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_mzeta/acb_mzeta_polylog.html (generated 2026-11-07).