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/*
Copyright (C) 2016 Pascal Molin
This file is part of Arb.
Arb is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "acb_dirichlet.h"
static void
gauss_sum_non_primitive(acb_t res, const dirichlet_group_t G, const dirichlet_char_t chi, ulong cond, slong prec)
{
slong k, mu = 1;
ulong NN0 = G->q / cond;
/* G(chi) = mu(N/N0)chi0(N/N0)G(chi0) */
if (NN0 % 2 == 0)
{
if (G->q % 4)
mu = -1;
else
{
acb_zero(res);
return;
}
}
for (k = G->neven; k < G->num; k++)
{
ulong p = G->P[k].p;
if (G->P[k].e > 1 && NN0 % (p*p) == 0)
{
acb_zero(res);
return;
}
if (NN0 % p == 0)
mu *= -1;
}
if (chi->n == 1)
{
acb_set_si(res, mu);
}
else
{
dirichlet_group_t G0;
dirichlet_char_t chi0;
acb_t z;
/* primitive char associated to chi */
dirichlet_subgroup_init(G0, G, cond);
dirichlet_char_init(chi0, G0);
dirichlet_char_lower(chi0, G0, chi, G);
acb_init(z);
acb_dirichlet_gauss_sum(z, G0, chi0, prec);
acb_dirichlet_chi(res, G0, chi0, NN0, prec);
acb_mul(res, res, z, prec);
acb_mul_si(res, res, mu, prec);
dirichlet_group_clear(G0);
dirichlet_char_clear(chi0);
acb_clear(z);
}
}
void
acb_dirichlet_gauss_sum(acb_t res, const dirichlet_group_t G, const dirichlet_char_t chi, slong prec)
{
ulong cond = dirichlet_conductor_char(G, chi);
/* TODO: no need, factor also does it... */
if (cond != G->q)
{
gauss_sum_non_primitive(res, G, chi, cond, prec);
}
else if (dirichlet_char_is_real(G, chi))
{
acb_dirichlet_gauss_sum_order2(res, G, chi, prec);
}
else if (G->num > 1 && G->num > G->neven)
{
acb_dirichlet_gauss_sum_factor(res, G, chi, prec);
}
else
{
/* must be non primitive */
if (acb_dirichlet_theta_length_d(G->q, 1, prec) > G->q)
acb_dirichlet_gauss_sum_naive(res, G, chi, prec);
else
acb_dirichlet_gauss_sum_theta(res, G, chi, prec);
}
}
void
acb_dirichlet_gauss_sum_ui(acb_t res, const dirichlet_group_t G, ulong a, slong prec)
{
dirichlet_char_t chi;
dirichlet_char_init(chi, G);
dirichlet_char_log(chi, G, a);
acb_dirichlet_gauss_sum(res, G, chi, prec);
dirichlet_char_clear(chi);
}
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