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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"
/* J_N(1,a) = sum on x = 1 mod some p | q */
ulong
jacobi_one_prime(ulong p, ulong e, ulong pe, ulong cond)
{
if (e > 1 && cond % (p*p) == 0)
{
return 0;
}
else
{
slong r = (e > 1) ? pe / p : 1;
if (cond % p)
return r * (p - 2);
else
return -r;
}
}
static ulong
jacobi_one(const dirichlet_group_t G, ulong cond)
{
slong k, r = 1;
for (k = 0; k < G->num; k++)
r *= jacobi_one_prime(G->P[k].p, G->P[k].e,
G->P[k].pe.n, cond);
return r;
}
void
acb_dirichlet_jacobi_sum(acb_t res, const dirichlet_group_t G, const dirichlet_char_t chi1, const dirichlet_char_t chi2, slong prec)
{
if (G->q_even > 1)
{
acb_zero(res);
}
else if (chi1->n == 1 || chi2->n == 1)
{
ulong cond = (chi1->n == 1) ? dirichlet_conductor_char(G, chi2) : dirichlet_conductor_char(G, chi1);
acb_set_si(res, jacobi_one(G, cond));
}
else if (nmod_mul(chi1->n, chi2->n, G->mod) == 1)
{
ulong n;
n = jacobi_one(G, dirichlet_conductor_char(G, chi1));
if (dirichlet_parity_char(G, chi1))
acb_set_si(res, -n);
else
acb_set_si(res, n);
}
else
{
if (G->q <= 150)
acb_dirichlet_jacobi_sum_naive(res, G, chi1, chi2, prec);
else if (G->num > 1)
acb_dirichlet_jacobi_sum_factor(res, G, chi1, chi2, prec);
else if (G->P[0].e > 1)
acb_dirichlet_jacobi_sum_naive(res, G, chi1, chi2, prec);
else
acb_dirichlet_jacobi_sum_gauss(res, G, chi1, chi2, prec);
}
}
void
acb_dirichlet_jacobi_sum_ui(acb_t res, const dirichlet_group_t G, ulong a, ulong b, slong prec)
{
if (G->q_even > 1)
{
acb_zero(res);
}
else if (a == 1 || b == 1)
{
ulong cond = (a == 1) ? dirichlet_conductor_ui(G, b) : dirichlet_conductor_ui(G, a);
acb_set_si(res, jacobi_one(G, cond));
}
else if (nmod_mul(a, b, G->mod) == 1)
{
ulong n;
n = jacobi_one(G, dirichlet_conductor_ui(G, a));
if (dirichlet_parity_ui(G, a))
acb_set_si(res, -n);
else
acb_set_si(res, n);
}
else
{
dirichlet_char_t chi1, chi2;
dirichlet_char_init(chi1, G);
dirichlet_char_init(chi2, G);
dirichlet_char_log(chi1, G, a);
dirichlet_char_log(chi2, G, b);
acb_dirichlet_jacobi_sum(res, G, chi1, chi2, prec);
dirichlet_char_clear(chi1);
dirichlet_char_clear(chi2);
}
}
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