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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"
void acb_zeta_si(acb_t z, slong s, slong prec);
static void
_acb_dirichlet_zeta(acb_t res, const acb_t s, slong prec)
{
acb_t a;
acb_init(a);
acb_one(a);
_acb_poly_zeta_cpx_series(res, s, a, 0, 1, prec);
acb_clear(a);
}
/* todo: use euler product for complex s, and check efficiency
for large negative integers */
void
acb_dirichlet_zeta(acb_t res, const acb_t s, slong prec)
{
double cutoff;
if (acb_is_int(s) &&
arf_cmpabs_2exp_si(arb_midref(acb_realref(s)), FLINT_BITS - 1) < 0)
{
acb_zeta_si(res, arf_get_si(arb_midref(acb_realref(s)), ARF_RND_DOWN), prec);
return;
}
if (arb_contains_zero(acb_imagref(s)) && arb_contains_si(acb_realref(s), 1))
{
acb_indeterminate(res);
return;
}
cutoff = 24.0 * prec * sqrt(prec);
if (arf_cmpabs_d(arb_midref(acb_imagref(s)), cutoff) >= 0 &&
arf_cmpabs_d(arb_midref(acb_realref(s)), 10 + prec * 0.1) <= 0)
{
acb_dirichlet_zeta_rs(res, s, 0, prec);
return;
}
if ((arf_sgn(arb_midref(acb_realref(s))) < 0) &&
!acb_contains_zero(s))
{
acb_t t, u, v;
slong wp = prec + 6;
acb_init(t);
acb_init(u);
acb_init(v);
acb_sub_ui(t, s, 1, wp);
/* 2 * (2pi)^(s-1) */
arb_const_pi(acb_realref(u), wp);
acb_mul_2exp_si(u, u, 1);
acb_pow(u, u, t, wp);
acb_mul_2exp_si(u, u, 1);
/* sin(pi*s/2) */
acb_mul_2exp_si(v, s, -1);
acb_sin_pi(v, v, wp);
acb_mul(u, u, v, wp);
/* gamma(1-s) zeta(1-s) */
acb_neg(t, t);
acb_gamma(v, t, wp);
acb_mul(u, u, v, wp);
_acb_dirichlet_zeta(v, t, prec);
acb_mul(res, u, v, prec);
acb_clear(t);
acb_clear(u);
acb_clear(v);
}
else
{
_acb_dirichlet_zeta(res, s, prec);
}
}
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