acb_dirichlet_l_si

Sets res to $L(s,\chi)$ for an integer $s$, using closed forms where they exist. An imprimitive character is reduced to the primitive character $\chi_0$ modulo $q_0$ inducing it, using $L(s,\chi) = L(s,\chi_0) \prod_{p \mid q} (1 - \chi_0(p) p^{-s})$, and $q_0 = 1$ gives the Riemann zeta function. For primitive $\chi$ modulo $q > 1$ with parity $\delta$ ($\chi(-1) = (-1)^{\delta}$):

• If $s = 1 - n \le 0$, then $L(s,\chi) = -B_{n,\chi}/n$ in terms of a generalized Bernoulli number, which is zero unless $n \equiv \delta \pmod 2$. (The functional equation and the Euler product are used instead when $n$ is large compared to the precision.)

• If $s \ge 1$ and $s \equiv \delta \pmod 2$, then

  $$L(s,\chi) = (-1)^{1 + (s - \delta)/2} \frac{\tau(\chi)}{2 i^{\delta}} \left(\frac{2 \pi}{q}\right)^s \frac{B_{s,\bar \chi}}{s!}$$

  where $\tau(\chi)$ is the Gauss sum (the Euler product is used instead for $s$ large compared to the precision). The generalized Bernoulli number $B_{s,\bar\chi} = q^{s-1} \sum_{a=1}^{q} \bar\chi(a) B_s(a/q)$ is computed with exact integer arithmetic grouped by the classes of residues with the same character value, so that only $\operatorname{ord}(\chi)$ roots of unity are needed; at high precision the cost is dominated by the Gauss sum, i.e. by about $q$ multiplications.

• If $s = 1$ and $\chi$ is even, then $L(1,\chi) = -(\tau(\chi)/q) \sum_{a=1}^{q-1} \bar\chi(a) \log|2 \sin(\pi a/q)|$, computed with one logarithm per class of residues with the same character value (a single logarithm for real $\chi$).
• Otherwise ($s \ge 2$ and $s \not\equiv \delta \pmod 2$, where no closed form is known), the algorithm is chosen among the Euler product (for $s$ large compared to the precision), Catalan's constant (arb_const_catalan, for $q = 4$, $s = 2$), acb_dirichlet_l_ui_borwein_bsplit (at high precision, for moderate $q$), the approximate functional equation (acb_dirichlet_l_fmpq_afe, for larger $q$) and the Hurwitz zeta function.