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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_dirichlet_hurwitz_precomp_bound(mag_t res, const acb_t s,
slong A, slong K, slong N)
{
acb_t s1;
mag_t x, t, TK, C;
slong sigmaK;
arf_t u;
if (A < 1 || K < 1 || N < 1)
{
mag_inf(res);
return;
}
/* sigmaK = re(s) + K, floor bound */
arf_init(u);
arf_set_mag(u, arb_radref(acb_realref(s)));
arf_sub(u, arb_midref(acb_realref(s)), u, MAG_BITS, ARF_RND_FLOOR);
arf_add_ui(u, u, K, MAG_BITS, ARF_RND_FLOOR);
if (arf_cmp_ui(u, 2) < 0 || arf_cmp_2exp_si(u, FLINT_BITS - 2) > 0)
{
mag_inf(res);
arf_clear(u);
return;
}
sigmaK = arf_get_si(u, ARF_RND_FLOOR);
arf_clear(u);
acb_init(s1);
mag_init(x);
mag_init(t);
mag_init(TK);
mag_init(C);
/* With N grid points, we will have |x| <= 1 / (2N). */
mag_one(x);
mag_div_ui(x, x, 2 * N);
/* T(K) = |x|^K |(s)_K| / K! * [1/A^(sigma+K) + ...] */
mag_pow_ui(TK, x, K);
acb_rising_ui_get_mag(t, s, K);
mag_mul(TK, TK, t);
mag_rfac_ui(t, K);
mag_mul(TK, TK, t);
/* Note: here we assume that mag_hurwitz_zeta_uiui uses an error bound
that is at least as large as the one used in the proof. */
mag_hurwitz_zeta_uiui(t, sigmaK, A);
mag_mul(TK, TK, t);
/* C = |x|/A (1 + 1/(K+sigma+A-1)) (1 + |s-1|/(K+1)) */
mag_div_ui(C, x, A);
mag_one(t);
mag_div_ui(t, t, sigmaK + A - 1);
mag_add_ui(t, t, 1);
mag_mul(C, C, t);
acb_sub_ui(s1, s, 1, MAG_BITS);
acb_get_mag(t, s1);
mag_div_ui(t, t, K + 1);
mag_add_ui(t, t, 1);
mag_mul(C, C, t);
mag_geom_series(t, C, 0);
mag_mul(res, TK, t);
acb_clear(s1);
mag_clear(x);
mag_clear(t);
mag_clear(TK);
mag_clear(C);
}
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