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https://github.com/vale981/arb
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90 lines
2.2 KiB
C
90 lines
2.2 KiB
C
/*
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Copyright (C) 2016 Fredrik Johansson
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This file is part of Arb.
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Arb is free software: you can redistribute it and/or modify it under
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the terms of the GNU Lesser General Public License (LGPL) as published
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by the Free Software Foundation; either version 2.1 of the License, or
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(at your option) any later version. See <http://www.gnu.org/licenses/>.
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*/
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#include "acb_dirichlet.h"
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/* Arias de Reyna, Theorem 4.2 */
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void
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acb_dirichlet_zeta_rs_bound(mag_t err, const acb_t s, slong K)
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{
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arb_t a;
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mag_t c1, c2, c3;
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slong e;
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if (!arb_is_positive(acb_imagref(s)) || K < 1 || !acb_is_finite(s))
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{
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mag_inf(err);
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return;
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}
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arb_init(a);
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arb_add_ui(a, acb_realref(s), K, MAG_BITS);
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arb_sub_ui(a, a, 2, MAG_BITS);
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if (!arb_is_nonnegative(acb_realref(s)) && !arb_is_nonnegative(a))
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{
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mag_inf(err);
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arb_clear(a);
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return;
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}
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mag_init(c1);
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mag_init(c2);
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mag_init(c3);
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/* c1 = 1/7 2^(3 sigma / 2) re(sigma) >= 0 */
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/* c1 < 1/2 re(sigma) < 0 */
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arf_set_mag(arb_midref(a), arb_radref(acb_realref(s)));
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arf_add(arb_midref(a), arb_midref(a), arb_midref(acb_realref(s)), MAG_BITS, ARF_RND_CEIL);
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if (arf_sgn(arb_midref(a)) <= 0)
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{
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mag_set_ui_2exp_si(c1, 1, -1);
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}
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else if (arf_cmp_2exp_si(arb_midref(a), FLINT_BITS - 4) < 0)
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{
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mag_one(c1);
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mag_div_ui(c1, c1, 7);
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e = arf_get_si(arb_midref(a), ARF_RND_CEIL);
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mag_mul_2exp_si(c1, c1, (3 * e + 1) / 2);
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if (mag_cmp_2exp_si(c1, -1) < 0)
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mag_set_ui_2exp_si(c1, 1, -1);
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}
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else
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{
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mag_inf(c1);
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}
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/* c2 = 1 / ((10/11) sqrt(t/(2pi))) = (11/10) sqrt((2pi)/t) */
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arb_get_mag_lower(c3, acb_imagref(s));
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mag_const_pi(c2);
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mag_mul_2exp_si(c2, c2, 1);
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mag_div(c2, c2, c3);
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mag_sqrt(c2, c2);
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mag_mul_ui(c2, c2, 11);
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mag_div_ui(c2, c2, 10);
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/* c2 = c2^(K+1) */
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mag_pow_ui(c2, c2, K + 1);
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/* c3 = gamma((K+1)/2) */
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mag_fac_ui(c3, K / 2);
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mag_mul(err, c1, c2);
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mag_mul(err, err, c3);
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mag_clear(c1);
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mag_clear(c2);
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mag_clear(c3);
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arb_clear(a);
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}
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