1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
/*
Copyright (C) 2016 Fredrik Johansson
This file is part of FLINT.
FLINT 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 3 of the License, or
(at your option) any later version. See <https://www.gnu.org/licenses/>.
*/
#include "acb.h"
#include "acb_dirichlet.h"
void
acb_dirichlet_eta(acb_t res, const acb_t s, slong prec)
{
if (!acb_is_finite(s))
{
acb_indeterminate(res);
}
else if (arb_contains_si(acb_realref(s), 1) && arb_contains_zero(acb_imagref(s)))
{
if (acb_is_one(s))
{
arb_const_log2(acb_realref(res), prec);
arb_zero(acb_imagref(res));
}
else
{
mag_t m;
int is_real = acb_is_real(s);
mag_init(m);
/* Taylor coefficients at s = 1 are bounded by |c_k| < 1/4^k. */
acb_sub_ui(res, s, 1, prec);
acb_get_mag(m, res);
mag_mul_2exp_si(m, m, -2);
mag_geom_series(m, m, 1);
arb_const_log2(acb_realref(res), prec);
arb_zero(acb_imagref(res));
if (is_real)
arb_add_error_mag(acb_realref(res), m);
else
acb_add_error_mag(res, m);
mag_clear(m);
}
}
else
{
acb_t t;
acb_init(t);
acb_one(t);
acb_mul_2exp_si(t, t, -1);
acb_pow(t, t, s, prec);
acb_mul_2exp_si(t, t, 1);
acb_sub_ui(t, t, 1, prec);
acb_neg(t, t);
acb_zeta(res, s, prec);
acb_mul(res, res, t, prec);
acb_clear(t);
}
}