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/*
Copyright (C) 2019 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
_mag_div_ui_ui(mag_t res, ulong a, ulong b)
{
mag_set_ui(res, a);
mag_div_ui(res, res, b);
}
void
acb_dirichlet_backlund_s_bound(mag_t res, const arb_t t)
{
if (!arb_is_nonnegative(t))
{
mag_inf(res);
}
else
{
mag_t m;
mag_init(m);
arb_get_mag(m, t);
if (mag_cmp_2exp_si(m, 8) < 0) /* 2^8 < 280 */
{
mag_one(res);
}
else if (mag_cmp_2exp_si(m, 22) < 0) /* 2^22 < 6.8*10^6 */
{
mag_set_ui(res, 2);
}
else if (mag_cmp_2exp_si(m, 29) < 0) /* 2^29 < 5.45*10^8 */
{
_mag_div_ui_ui(res, 231366, 100000);
}
else
{
/* |S(t)| <= 0.112*log(t) + 0.278*log(log(t)) + 2.51 */
mag_t c, logm;
mag_init(c);
mag_init(logm);
mag_log(logm, m);
_mag_div_ui_ui(c, 278, 1000);
mag_log(res, logm);
mag_mul(res, res, c);
_mag_div_ui_ui(c, 112, 1000);
mag_addmul(res, c, logm);
_mag_div_ui_ui(c, 251, 100);
mag_add(res, res, c);
mag_clear(c);
mag_clear(logm);
}
mag_clear(m);
}
}
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