/* assumes no aliasing */ slong acb_lambertw_initial(acb_t res, const acb_t z, const acb_t ez1, const fmpz_t k, slong prec) { /* Handle z very close to 0 on the principal branch. */ if (fmpz_is_zero(k) && (arf_cmpabs_2exp_si(arb_midref(acb_realref(z)), -20) <= 0 && arf_cmpabs_2exp_si(arb_midref(acb_imagref(z)), -20) <= 0)) { acb_set(res, z); acb_submul(res, res, res, prec); return 40; /* could be tightened... */ } /* For moderate input not close to the branch point, compute a double approximation as the initial value. */ if (fmpz_is_zero(k) && arf_cmpabs_2exp_si(arb_midref(acb_realref(z)), 400) < 0 && arf_cmpabs_2exp_si(arb_midref(acb_imagref(z)), 400) < 0 && (arf_cmp_d(arb_midref(acb_realref(z)), -0.37) < 0 || arf_cmp_d(arb_midref(acb_realref(z)), -0.36) > 0 || arf_cmpabs_d(arb_midref(acb_imagref(z)), 0.01) > 0)) { acb_lambertw_principal_d(res, z); return 48; } /* Check if we are close to the branch point at -1/e. */ if ((fmpz_is_zero(k) || (fmpz_is_one(k) && arb_is_negative(acb_imagref(z))) || (fmpz_equal_si(k, -1) && arb_is_nonnegative(acb_imagref(z)))) && ((arf_cmpabs_2exp_si(arb_midref(acb_realref(ez1)), -2) <= 0 && arf_cmpabs_2exp_si(arb_midref(acb_imagref(ez1)), -2) <= 0))) { acb_t t; acb_init(t); acb_mul_2exp_si(t, ez1, 1); mag_zero(arb_radref(acb_realref(t))); mag_zero(arb_radref(acb_imagref(t))); acb_mul_ui(t, t, 3, prec); acb_sqrt(t, t, prec); if (!fmpz_is_zero(k)) acb_neg(t, t); acb_lambertw_branchpoint_series(res, t, 0, prec); acb_clear(t); return 1; /* todo: estimate */ } acb_lambertw_initial_asymp(res, z, k, prec); return 1; /* todo: estimate */ }
/* todo: use euler product for complex s, and check efficiency for large negative integers */ void acb_dirichlet_zeta(acb_t res, const acb_t s, slong prec) { acb_t a; double cutoff; if (acb_is_int(s) && arf_cmpabs_2exp_si(arb_midref(acb_realref(s)), FLINT_BITS - 1) < 0) { acb_zeta_si(res, arf_get_si(arb_midref(acb_realref(s)), ARF_RND_DOWN), prec); return; } cutoff = 24.0 * prec * sqrt(prec); if (arf_cmpabs_d(arb_midref(acb_imagref(s)), cutoff) >= 0 && arf_cmpabs_d(arb_midref(acb_realref(s)), 10 + prec * 0.1) <= 0) { acb_dirichlet_zeta_rs(res, s, 0, prec); return; } acb_init(a); acb_one(a); if (arf_sgn(arb_midref(acb_realref(s))) < 0) { acb_t t, u, v; slong wp = prec + 6; acb_init(t); acb_init(u); acb_init(v); acb_sub_ui(t, s, 1, wp); /* 2 * (2pi)^(s-1) */ arb_const_pi(acb_realref(u), wp); acb_mul_2exp_si(u, u, 1); acb_pow(u, u, t, wp); acb_mul_2exp_si(u, u, 1); /* sin(pi*s/2) */ acb_mul_2exp_si(v, s, -1); acb_sin_pi(v, v, wp); acb_mul(u, u, v, wp); /* gamma(1-s) zeta(1-s) */ acb_neg(t, t); acb_gamma(v, t, wp); acb_mul(u, u, v, wp); acb_hurwitz_zeta(v, t, a, wp); acb_mul(res, u, v, prec); acb_clear(t); acb_clear(u); acb_clear(v); } else { acb_hurwitz_zeta(res, s, a, prec); } acb_clear(a); }
void acb_dirichlet_l(acb_t res, const acb_t s, const dirichlet_group_t G, const dirichlet_char_t chi, slong prec) { if (!acb_is_finite(s)) { acb_indeterminate(res); } else if (G == NULL || G->q == 1) { acb_dirichlet_zeta(res, s, prec); } else if (dirichlet_char_is_primitive(G, chi) && (arf_cmp_d(arb_midref(acb_realref(s)), -0.5) < 0 || (G->q != 1 && dirichlet_parity_char(G, chi) == 0 && arf_cmpabs_d(arb_midref(acb_imagref(s)), 0.125) < 0 && arf_cmp_d(arb_midref(acb_realref(s)), 0.125) < 0))) { /* use functional equation */ acb_t t, u, v; int parity; ulong q; parity = dirichlet_parity_char(G, chi); q = G->q; acb_init(t); acb_init(u); acb_init(v); /* gamma((1-s+p)/2) / gamma((s+p)/2) */ acb_add_ui(t, s, parity, prec); acb_mul_2exp_si(t, t, -1); acb_rgamma(t, t, prec); if (!acb_is_zero(t)) /* assumes q != 1 when s = 0 */ { acb_neg(u, s); acb_add_ui(u, u, 1 + parity, prec); acb_mul_2exp_si(u, u, -1); acb_gamma(u, u, prec); acb_mul(t, t, u, prec); /* epsilon */ acb_dirichlet_root_number(u, G, chi, prec); acb_mul(t, t, u, prec); /* (pi/q)^(s-1/2) */ acb_const_pi(u, prec); acb_div_ui(u, u, q, prec); acb_set_d(v, -0.5); acb_add(v, v, s, prec); acb_pow(u, u, v, prec); acb_mul(t, t, u, prec); acb_sub_ui(u, s, 1, prec); acb_neg(u, u); acb_conj(u, u); acb_dirichlet_l_general(u, u, G, chi, prec); acb_conj(u, u); acb_mul(t, t, u, prec); if (dirichlet_char_is_real(G, chi) && acb_is_real(s)) arb_zero(acb_imagref(t)); } acb_set(res, t); acb_clear(t); acb_clear(u); acb_clear(v); } else { acb_dirichlet_l_general(res, s, G, chi, prec); } }