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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"
/* Arias de Reyna, Theorem 4.2 */
void
acb_dirichlet_zeta_rs_bound(mag_t err, const acb_t s, slong K)
{
arb_t a;
mag_t c1, c2, c3;
slong e;
if (!arb_is_positive(acb_imagref(s)) || K < 1 || !acb_is_finite(s))
{
mag_inf(err);
return;
}
arb_init(a);
arb_add_ui(a, acb_realref(s), K, MAG_BITS);
arb_sub_ui(a, a, 2, MAG_BITS);
if (!arb_is_nonnegative(acb_realref(s)) && !arb_is_nonnegative(a))
{
mag_inf(err);
arb_clear(a);
return;
}
mag_init(c1);
mag_init(c2);
mag_init(c3);
/* c1 = 1/7 2^(3 sigma / 2) re(sigma) >= 0 */
/* c1 < 1/2 re(sigma) < 0 */
arf_set_mag(arb_midref(a), arb_radref(acb_realref(s)));
arf_add(arb_midref(a), arb_midref(a), arb_midref(acb_realref(s)), MAG_BITS, ARF_RND_CEIL);
if (arf_sgn(arb_midref(a)) <= 0)
{
mag_set_ui_2exp_si(c1, 1, -1);
}
else if (arf_cmp_2exp_si(arb_midref(a), FLINT_BITS - 4) < 0)
{
mag_one(c1);
mag_div_ui(c1, c1, 7);
e = arf_get_si(arb_midref(a), ARF_RND_CEIL);
mag_mul_2exp_si(c1, c1, (3 * e + 1) / 2);
if (mag_cmp_2exp_si(c1, -1) < 0)
mag_set_ui_2exp_si(c1, 1, -1);
}
else
{
mag_inf(c1);
}
/* c2 = 1 / ((10/11) sqrt(t/(2pi))) = (11/10) sqrt((2pi)/t) */
arb_get_mag_lower(c3, acb_imagref(s));
mag_const_pi(c2);
mag_mul_2exp_si(c2, c2, 1);
mag_div(c2, c2, c3);
mag_sqrt(c2, c2);
mag_mul_ui(c2, c2, 11);
mag_div_ui(c2, c2, 10);
/* c2 = c2^(K+1) */
mag_pow_ui(c2, c2, K + 1);
/* c3 = gamma((K+1)/2) */
mag_fac_ui(c3, K / 2);
mag_mul(err, c1, c2);
mag_mul(err, err, c3);
mag_clear(c1);
mag_clear(c2);
mag_clear(c3);
arb_clear(a);
}
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