acb_mzeta_epstein_shell, acb_mzeta_epstein_gram_shell

The algorithm by shells, for exact rational input: the first form requires $A$, $x$, $y$ to be exact (dyadic, so that the Gram matrix $A^T A$ and the coordinates are rational); the second takes the Gram matrix $G$ of the lattice and the coordinates $u$, $v$ with $x = A u$, $y = A^{-T} v$ (then $|x - z|^2 = (u - n)^T G (u - n)$ and $y \cdot z = v \cdot n$ for $z = A n$), so that lattices such as the hexagonal one can be given exactly. $\nu$ must be real and exact with a short dyadic value. Returns 0 (not applicable) for the regularised function, for $d > 8$, for denominators of $u$, $v$ above 64 and when the estimated cost of the shell counts is too large.

The lattice points are grouped into shells of equal norm, counted exactly (by products of theta series of low-rank blocks, by enumeration, or from sums of three squares), so that each side of the Crandall splitting needs one incomplete gamma value per shell; the splitting parameter is chosen so that one side has exact rational arguments (binary splitting) and, for integer indices such as the Coulomb case $\nu = d - 2$, the other side is a power series with rational coefficients. See Sums by shells. The computation is parallel with up to flint_get_num_threads() threads, and the result does not depend on the number of threads.