void _acb_dirichlet_theta_arb_series(acb_t res, const dirichlet_group_t G, const dirichlet_char_t chi, const arb_t xt, slong len, slong prec) { acb_ptr a; a = _acb_vec_init(len); acb_dirichlet_chi_vec(a, G, chi, len, prec); if (dirichlet_parity_char(G, chi)) { slong k; for (k = 2; k < len; k++) acb_mul_si(a + k, a + k, k, prec); } acb_dirichlet_qseries_arb(res, a, xt, len, prec); _acb_vec_clear(a, len); }
void acb_dirichlet_vec_mellin_arb(acb_ptr res, const dirichlet_group_t G, const dirichlet_char_t chi, slong len, const arb_t t, slong n, slong prec) { slong k; arb_t tk, xt, stk, st; acb_ptr a; mag_t e; a = _acb_vec_init(len); acb_dirichlet_chi_vec(a, G, chi, len, prec); if (dirichlet_parity_char(G, chi)) { for (k = 2; k < len; k++) acb_mul_si(a + k, a + k, k, prec); } arb_init(tk); arb_init(xt); arb_init(st); arb_init(stk); mag_init(e); arb_sqrt(st, t, prec); arb_one(tk); arb_one(stk); for (k = 0; k < n; k++) { _acb_dirichlet_theta_argument_at_arb(xt, G->q, tk, prec); mag_tail_kexpk2_arb(e, xt, len); arb_neg(xt, xt); arb_exp(xt, xt, prec); /* TODO: reduce len */ acb_dirichlet_qseries_arb(res + k, a, xt, len, prec); acb_add_error_mag(res + k, e); acb_mul_arb(res + k, res + k, stk, prec); arb_mul(tk, tk, t, prec); arb_mul(stk, stk, st, prec); } mag_clear(e); arb_clear(xt); arb_clear(tk); arb_clear(stk); arb_clear(st); _acb_vec_clear(a, len); }
void acb_modular_eisenstein(acb_ptr r, const acb_t tau, slong len, slong prec) { psl2z_t g; arf_t one_minus_eps; acb_t tau_prime, t1, t2, t3, t4, q; slong m, n; if (len < 1) return; psl2z_init(g); arf_init(one_minus_eps); acb_init(tau_prime); acb_init(t1); acb_init(t2); acb_init(t3); acb_init(t4); acb_init(q); arf_set_ui_2exp_si(one_minus_eps, 63, -6); acb_modular_fundamental_domain_approx(tau_prime, g, tau, one_minus_eps, prec); acb_exp_pi_i(q, tau_prime, prec); acb_modular_theta_const_sum(t2, t3, t4, q, prec); /* fourth powers of the theta functions (a, b, c) */ acb_mul(t2, t2, t2, prec); acb_mul(t2, t2, t2, prec); acb_mul(t2, t2, q, prec); acb_mul(t3, t3, t3, prec); acb_mul(t3, t3, t3, prec); acb_mul(t4, t4, t4, prec); acb_mul(t4, t4, t4, prec); /* c2 = pi^4 * (a^8 + b^8 + c^8) / 30 */ /* c3 = pi^6 * (b^12 + c^12 - 3a^8 * (b^4+c^4)) / 180 */ /* r = a^8 */ acb_mul(r, t2, t2, prec); if (len > 1) { /* r[1] = -3 a^8 * (b^4 + c^4) */ acb_add(r + 1, t3, t4, prec); acb_mul(r + 1, r + 1, r, prec); acb_mul_si(r + 1, r + 1, -3, prec); } /* b^8 */ acb_mul(t1, t3, t3, prec); acb_add(r, r, t1, prec); /* b^12 */ if (len > 1) acb_addmul(r + 1, t1, t3, prec); /* c^8 */ acb_mul(t1, t4, t4, prec); acb_add(r, r, t1, prec); /* c^12 */ if (len > 1) acb_addmul(r + 1, t1, t4, prec); acb_const_pi(t1, prec); acb_mul(t1, t1, t1, prec); acb_mul(t2, t1, t1, prec); acb_mul(r, r, t2, prec); acb_div_ui(r, r, 30, prec); if (len > 1) { acb_mul(t2, t2, t1, prec); acb_mul(r + 1, r + 1, t2, prec); acb_div_ui(r + 1, r + 1, 189, prec); } /* apply modular transformation */ if (!fmpz_is_zero(&g->c)) { acb_mul_fmpz(t1, tau, &g->c, prec); acb_add_fmpz(t1, t1, &g->d, prec); acb_inv(t1, t1, prec); acb_mul(t1, t1, t1, prec); acb_mul(t2, t1, t1, prec); acb_mul(r, r, t2, prec); if (len > 1) { acb_mul(t2, t1, t2, prec); acb_mul(r + 1, r + 1, t2, prec); } } /* compute more coefficients using recurrence */ for (n = 4; n < len + 2; n++) { acb_zero(r + n - 2); m = 2; for (m = 2; m * 2 < n; m++) acb_addmul(r + n - 2, r + m - 2, r + n - m - 2, prec); acb_mul_2exp_si(r + n - 2, r + n - 2, 1); if (n % 2 == 0) acb_addmul(r + n - 2, r + n / 2 - 2, r + n / 2 - 2, prec); acb_mul_ui(r + n - 2, r + n - 2, 3, prec); acb_div_ui(r + n - 2, r + n - 2, (2 * n + 1) * (n - 3), prec); } /* convert c's to G's */ for (n = 0; n < len; n++) acb_div_ui(r + n, r + n, 2 * n + 3, prec); psl2z_clear(g); arf_clear(one_minus_eps); acb_clear(tau_prime); acb_clear(t1); acb_clear(t2); acb_clear(t3); acb_clear(t4); acb_clear(q); }
int main() { long iter; flint_rand_t state; printf("lambda...."); fflush(stdout); flint_randinit(state); for (iter = 0; iter < 10000; iter++) { acb_t tau1, tau2, z1, z2, z3, t; long e0, prec0, prec1, prec2, step; acb_init(tau1); acb_init(tau2); acb_init(z1); acb_init(z2); acb_init(z3); acb_init(t); e0 = 1 + n_randint(state, 100); prec0 = 2 + n_randint(state, 1000); prec1 = 2 + n_randint(state, 1000); prec2 = 2 + n_randint(state, 1000); acb_randtest(tau1, state, prec0, e0); acb_randtest(tau2, state, prec0, e0); acb_randtest(z1, state, prec0, e0); acb_randtest(z2, state, prec0, e0); acb_set(tau2, tau1); step = n_randint(state, 10); /* Test invariance */ while (step --> 0) { if (n_randint(state, 2)) { acb_add_ui(tau2, tau2, 2, prec0); } else { acb_mul_si(z1, tau2, -2, prec0); acb_add_ui(z1, z1, 1, prec0); acb_div(tau2, tau2, z1, prec0); } } acb_modular_lambda(z1, tau1, prec1); acb_modular_lambda(z2, tau2, prec2); /* Compare with eta */ acb_mul_2exp_si(tau1, tau1, -1); acb_modular_eta(z3, tau1, prec2); acb_mul_2exp_si(tau1, tau1, 2); acb_modular_eta(t, tau1, prec2); acb_mul(t, t, t, prec2); acb_mul(z3, z3, t, prec2); acb_mul_2exp_si(tau1, tau1, -1); acb_modular_eta(t, tau1, prec2); acb_pow_ui(t, t, 3, prec2); acb_div(z3, z3, t, prec2); acb_pow_ui(z3, z3, 8, prec2); acb_mul_2exp_si(z3, z3, 4); if (!acb_overlaps(z1, z2) || !acb_overlaps(z1, z3)) { printf("FAIL (overlap)\n"); printf("tau1 = "); acb_printd(tau1, 15); printf("\n\n"); printf("tau2 = "); acb_printd(tau2, 15); printf("\n\n"); printf("z1 = "); acb_printd(z1, 15); printf("\n\n"); printf("z2 = "); acb_printd(z2, 15); printf("\n\n"); printf("z3 = "); acb_printd(z3, 15); printf("\n\n"); abort(); } acb_modular_lambda(tau1, tau1, prec2); if (!acb_overlaps(z1, tau1)) { printf("FAIL (aliasing)\n"); printf("tau1 = "); acb_printd(tau1, 15); printf("\n\n"); printf("tau2 = "); acb_printd(tau2, 15); printf("\n\n"); printf("z1 = "); acb_printd(z1, 15); printf("\n\n"); printf("z2 = "); acb_printd(z2, 15); printf("\n\n"); abort(); } acb_clear(tau1); acb_clear(tau2); acb_clear(z1); acb_clear(z2); acb_clear(z3); acb_clear(t); } flint_randclear(state); flint_cleanup(); printf("PASS\n"); return EXIT_SUCCESS; }