acb_dirichlet_l_fmpq_afe_jet

Sets res to the Taylor coefficients $L(s), L'(s), \ldots, L^{(\text{len}-1)}(s)/(\text{len}-1)!$ for rational $s$, with deflate as in acb_dirichlet_l_jet. This uses the same approximate functional equation and precision management as acb_dirichlet_l_fmpq_afe, with every quantity replaced by a power series in $x$ where $s \to s + x$. The required expansions of the incomplete gamma functions $\Gamma(a + x/2, \pi n^2 / q)$ with respect to the parameter are computed by jet versions of the binary splitting and bit-burst algorithms (see _arb_gamma_upper_fmpq_step_bsplit_jet), with truncation errors bounded separately for each coefficient. For fixed len, the cost is $\tilde O(p^{1.5})$ at precision $p$, roughly len times the cost of acb_dirichlet_l_fmpq_afe; this makes it possible to compute, for example, Glaisher's constant (via $\zeta'(-1)$), Stieltjes constants and their generalizations to Dirichlet $L$-functions, or the second-order Landau-Ramanujan constant (via $\zeta'(s)$ and $L'(s, \chi_{-4})$ at $s = 2^k$) in quasi-$p^{1.5}$ time. An imprimitive character $\chi$ modulo $q$ induced by $\chi_0$ is handled by evaluating $L(s + x, \chi_0)$ and multiplying by the Euler factors $\prod_{p \mid q} (1 - \chi_0(p) p^{-s-x})$. The deflation only has an effect for $q = 1$ (as in acb_dirichlet_zeta_jet); for the Riemann zeta function at $s = 1$, deflate must be set, and for a principal character modulo $q > 1$, deflate must not be set; otherwise the output is indeterminate.

The terms of the approximate functional equation are independent, and are computed in parallel when FLINT is allowed to use several threads (see flint_set_num_threads). The terms are added up in a fixed order, so the output does not depend on the number of threads. Each term is added as soon as all earlier terms are done; a term that finishes early waits in a small buffer (a few terms per thread), so the memory usage is $O(t \cdot \text{len} \cdot p)$ bits with $t$ threads, independent of the number of terms $O(\sqrt{q p})$. The same applies to acb_dirichlet_l_fmpq_afe, which is the case len = 1 of this function.