mirror of
https://github.com/vale981/arb
synced 2025-03-06 18:01:39 -05:00
114 lines
2.6 KiB
C
114 lines
2.6 KiB
C
/*
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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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static void
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gauss_sum_non_primitive(acb_t res, const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi, slong prec)
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{
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slong k, mu = 1;
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ulong NN0 = G->q / chi->conductor;
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/* G(chi) = mu(N/N0)chi0(N/N0)G(chi0) */
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if (NN0 % 2 == 0)
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{
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if (G->q % 4)
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mu = -1;
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else
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{
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acb_zero(res);
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return;
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}
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}
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for (k = G->neven; k < G->num; k++)
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{
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ulong p = G->P[k].p;
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if (G->P[k].e > 1 && NN0 % (p*p) == 0)
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{
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acb_zero(res);
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return;
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}
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if (NN0 % p == 0)
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mu *= -1;
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}
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if (chi->x->n == 1)
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{
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acb_set_si(res, mu);
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}
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else
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{
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acb_dirichlet_group_t G0;
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acb_dirichlet_char_t chi0;
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acb_t z;
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acb_dirichlet_subgroup_init(G0, G, chi->conductor);
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acb_dirichlet_char_init(chi0, G);
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acb_dirichlet_char_primitive(chi0, G0, G, chi);
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acb_init(z);
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acb_dirichlet_gauss_sum(z, G0, chi0, prec);
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acb_dirichlet_chi(res, G0, chi0, NN0, prec);
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acb_mul(res, res, z, prec);
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acb_mul_si(res, res, mu, prec);
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acb_dirichlet_group_clear(G0);
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acb_dirichlet_char_clear(chi0);
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acb_clear(z);
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}
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}
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void
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acb_dirichlet_gauss_sum_order2(acb_t res, const acb_dirichlet_char_t chi, slong prec)
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{
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if (chi->parity)
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{
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arb_zero(acb_realref(res));
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arb_sqrt_ui(acb_imagref(res), chi->q, prec);
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}
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else
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{
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arb_zero(acb_imagref(res));
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arb_sqrt_ui(acb_realref(res), chi->q, prec);
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}
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}
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void
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acb_dirichlet_gauss_sum(acb_t res, const acb_dirichlet_group_t G, const acb_dirichlet_char_t chi, slong prec)
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{
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/* TODO: no need, factor also does it... */
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if (chi->conductor != G->q)
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{
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gauss_sum_non_primitive(res, G, chi, prec);
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}
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else if (chi->order.n <= 2)
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{
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acb_dirichlet_gauss_sum_order2(res, chi, prec);
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}
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else if (G->num > 1 && G->num > G->neven)
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{
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acb_dirichlet_gauss_sum_factor(res, G, chi, prec);
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}
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else
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{
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if (acb_dirichlet_theta_length_d(G->q, 1, prec) > G->q)
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acb_dirichlet_gauss_sum_naive(res, G, chi, prec);
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else
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acb_dirichlet_gauss_sum_theta(res, G, chi, prec);
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}
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}
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