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/*
Copyright (C) 2016 Fredrik Johansson
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"
#include "acb_poly.h"
void
acb_dirichlet_hardy_theta(acb_ptr res, const acb_t t,
const dirichlet_group_t G, const dirichlet_char_t chi,
slong len, slong prec)
{
acb_struct y[2];
arb_t c;
slong k;
int parity;
ulong q;
if (len <= 0)
return;
if (t == res)
{
acb_init(y);
acb_set(y, t);
acb_dirichlet_hardy_theta(res, y, G, chi, len, prec);
acb_clear(y);
return;
}
if (G == NULL)
{
parity = 0;
q = 1;
}
else
{
parity = dirichlet_parity_char(G, chi);
q = G->q;
if (q != dirichlet_conductor_char(G, chi))
{
flint_printf("hardy theta: need primitive character\n");
flint_abort();
}
}
if (!acb_is_finite(t))
{
_acb_vec_indeterminate(res, len);
return;
}
acb_init(y + 0);
acb_init(y + 1);
arb_init(c);
/* res = log gamma((s+parity)/2), s = 0.5+i(t+x) */
acb_mul_onei(y, t);
arb_set_d(c, 0.5 + parity);
arb_add(acb_realref(y), acb_realref(y), c, prec);
acb_mul_2exp_si(y, y, -1);
acb_onei(y + 1);
acb_mul_2exp_si(y + 1, y + 1, -1);
_acb_poly_lgamma_series(res, y, FLINT_MIN(len, 2), len, prec);
if (acb_is_real(t))
{
for (k = 0; k < len; k++)
{
arb_set(acb_realref(res + k), acb_imagref(res + k));
arb_zero(acb_imagref(res + k));
}
}
else
{
acb_ptr v = _acb_vec_init(len);
/* v = log gamma(((1-s)+parity)/2), s = 0.5+i(t+x) */
acb_div_onei(y, t);
arb_set_d(c, 0.5 + parity);
arb_add(acb_realref(y), acb_realref(y), c, prec);
acb_mul_2exp_si(y, y, -1);
acb_neg(y + 1, y + 1);
_acb_poly_lgamma_series(v, y, FLINT_MIN(len, 2), len, prec);
_acb_vec_sub(res, res, v, len, prec);
for (k = 0; k < len; k++)
{
acb_div_onei(res + k, res + k);
acb_mul_2exp_si(res + k, res + k, -1);
}
_acb_vec_clear(v, len);
}
/* (t+x) [-(1/2) log(pi/q)] */
arb_const_pi(c, prec);
arb_div_ui(c, c, q, prec);
arb_log(c, c, prec);
arb_mul_2exp_si(c, c, -1);
acb_submul_arb(res, t, c, prec);
if (len > 1)
acb_sub_arb(res + 1, res + 1, c, prec);
/* i log(eps) / 2 */
if (q > 1)
{
acb_dirichlet_root_number(y, G, chi, prec);
acb_arg(c, y, prec);
arb_mul_2exp_si(c, c, -1);
arb_sub(acb_realref(res), acb_realref(res), c, prec);
}
acb_clear(y + 0);
acb_clear(y + 1);
arb_clear(c);
}
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