mirror of
https://github.com/vale981/arb
synced 2025-03-05 09:21:38 -05:00
111 lines
3.2 KiB
C
111 lines
3.2 KiB
C
/*
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Copyright (C) 2016 Fredrik Johansson
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Copyright (C) 2016 Pascal Molin
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This file is part of Arb.
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Arb is free software: you can redistribute it and/or modify it under
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the terms of the GNU Lesser General Public License (LGPL) as published
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by the Free Software Foundation; either version 2.1 of the License, or
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(at your option) any later version. See <http://www.gnu.org/licenses/>.
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*/
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#include "acb_dirichlet.h"
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void
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acb_dirichlet_l_general(acb_t res, const acb_t s,
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const dirichlet_group_t G, const dirichlet_char_t chi, slong prec)
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{
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/* this cutoff is probably too conservative when q is large */
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if (arf_cmp_d(arb_midref(acb_realref(s)), 8 + 0.5 * prec / log(prec)) >= 0)
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{
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acb_dirichlet_l_euler_product(res, s, G, chi, prec);
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}
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else
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{
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slong wp = prec + n_clog(G->phi_q, 2);
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acb_dirichlet_hurwitz_precomp_t pre;
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acb_dirichlet_hurwitz_precomp_init_num(pre, s, acb_is_one(s), G->phi_q, wp);
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acb_dirichlet_l_hurwitz(res, s, pre, G, chi, prec);
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acb_dirichlet_hurwitz_precomp_clear(pre);
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}
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}
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void
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acb_dirichlet_l(acb_t res, const acb_t s,
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const dirichlet_group_t G, const dirichlet_char_t chi, slong prec)
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{
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if (!acb_is_finite(s))
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{
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acb_indeterminate(res);
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}
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else if (G == NULL || G->q == 1)
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{
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acb_dirichlet_zeta(res, s, prec);
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}
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else if (dirichlet_char_is_primitive(G, chi) &&
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(arf_cmp_d(arb_midref(acb_realref(s)), -0.5) < 0 ||
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(G->q != 1 && dirichlet_parity_char(G, chi) == 0 &&
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arf_cmpabs_d(arb_midref(acb_imagref(s)), 0.125) < 0 &&
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arf_cmp_d(arb_midref(acb_realref(s)), 0.125) < 0)))
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{
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/* use functional equation */
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acb_t t, u, v;
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int parity;
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ulong q;
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parity = dirichlet_parity_char(G, chi);
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q = G->q;
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acb_init(t);
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acb_init(u);
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acb_init(v);
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/* gamma((1-s+p)/2) / gamma((s+p)/2) */
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acb_add_ui(t, s, parity, prec);
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acb_mul_2exp_si(t, t, -1);
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acb_rgamma(t, t, prec);
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if (!acb_is_zero(t)) /* assumes q != 1 when s = 0 */
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{
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acb_neg(u, s);
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acb_add_ui(u, u, 1 + parity, prec);
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acb_mul_2exp_si(u, u, -1);
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acb_gamma(u, u, prec);
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acb_mul(t, t, u, prec);
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/* epsilon */
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acb_dirichlet_root_number(u, G, chi, prec);
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acb_mul(t, t, u, prec);
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/* (pi/q)^(s-1/2) */
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acb_const_pi(u, prec);
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acb_div_ui(u, u, q, prec);
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acb_set_d(v, -0.5);
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acb_add(v, v, s, prec);
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acb_pow(u, u, v, prec);
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acb_mul(t, t, u, prec);
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acb_sub_ui(u, s, 1, prec);
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acb_neg(u, u);
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acb_conj(u, u);
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acb_dirichlet_l_general(u, u, G, chi, prec);
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acb_conj(u, u);
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acb_mul(t, t, u, prec);
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if (dirichlet_char_is_real(G, chi) && acb_is_real(s))
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arb_zero(acb_imagref(t));
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}
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acb_set(res, t);
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acb_clear(t);
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acb_clear(u);
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acb_clear(v);
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}
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else
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{
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acb_dirichlet_l_general(res, s, G, chi, prec);
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}
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}
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