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/*
Copyright (C) 2018 D.H.J Polymath
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
_acb_dirichlet_xi(acb_t res, const acb_t s, slong prec)
{
acb_t pi, z1, z2, z3;
acb_init(pi);
acb_init(z1);
acb_init(z2);
acb_init(z3);
/* (s-1) * pi^(-s/2) * gamma(1 + s/2) * zeta(s) */
acb_const_pi(pi, prec);
acb_mul_2exp_si(z1, s, -1);
acb_neg(z1, z1);
acb_pow(z1, pi, z1, prec);
acb_mul_2exp_si(z2, s, -1);
acb_add_ui(z2, z2, 1, prec);
acb_gamma(z2, z2, prec);
acb_zeta(z3, s, prec);
acb_sub_ui(res, s, 1, prec);
acb_mul(res, res, z1, prec);
acb_mul(res, res, z2, prec);
acb_mul(res, res, z3, prec);
acb_clear(pi);
acb_clear(z1);
acb_clear(z2);
acb_clear(z3);
}
void acb_dirichlet_xi(acb_t res, const acb_t s, slong prec)
{
if (!acb_is_finite(s))
{
acb_indeterminate(res);
}
else if (acb_is_one(s))
{
acb_one(res);
acb_mul_2exp_si(res, res, -1);
}
else if ((arf_sgn(arb_midref(acb_realref(s))) < 0 &&
!acb_contains_zero(s)) ||
(arb_contains_si(acb_realref(s), 1) && /* also intervals around s = 1 */
arb_contains_zero(acb_imagref(s))))
{
acb_sub_ui(res, s, 1, prec);
acb_neg(res, res);
_acb_dirichlet_xi(res, res, prec);
}
else
{
_acb_dirichlet_xi(res, s, prec);
}
}
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