acb_dirichlet_l_ui_borwein_bsplit

Sets res to $L(s,\chi)$ for an integer $s \ge 1$ and a primitive character $\chi$ modulo $q$ (with G = NULL for $q = 1$), using a generalization of Borwein's algorithm for the Riemann zeta function (arb_zeta_ui_borwein_bsplit) with binary splitting, which costs $\tilde O(\phi(q) p)$ at precision $p$.

If $4 \mid q$, the 2-component of $\chi$ is primitive, which implies that $\chi(n + Q) = -\chi(n)$ with $Q = q/2$; if $q$ is odd, we instead use $\sum_n (-1)^{n-1} \chi(n) n^{-s} = (1 - \chi(2) 2^{1-s}) L(s,\chi)$ with $Q = q$. In both cases, $L(s,\chi)$ is a linear combination over the residues $a$ modulo $Q$ of alternating series $\sum_{j \ge 0} (-1)^j (Q j + a)^{-s}$, whose terms are moments of a positive measure on $[0,1]$ with total mass $a^{-s}$. Borwein's acceleration with $n$ terms therefore has error at most $3 a^{-s} / (3+\sqrt{8})^n$ for each series. The series are evaluated together by one binary splitting over vectors (several in parallel when threads are available), and the character values only enter in the final linear combination.

The output is indeterminate if the character is not primitive, if $q \equiv 2 \pmod 4$ (there are then no primitive characters), and if $s = 1$, $q$ is odd and $\chi(2) = 1$ (when the factor $1 - \chi(2) 2^{1-s}$ vanishes).