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
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
|
/*
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_zeta_rs_r(acb_t res, const acb_t s, slong K, slong prec)
{
arb_ptr dk, pipow;
acb_ptr Fp;
arb_t a, p, api2, api2pow;
acb_t U, S, u, v;
fmpz_t N;
mag_t err;
slong j, k, wp, K_limit;
/* determinate K automatically */
if (K <= 0)
{
double sigma, t, log2err, best_log2err;
slong best_K;
sigma = arf_get_d(arb_midref(acb_realref(s)), ARF_RND_DOWN);
t = arf_get_d(arb_midref(acb_imagref(s)), ARF_RND_DOWN);
if (!(sigma > -1e6 && sigma < 1e6) || !(t > 1 && t < 1e40))
{
acb_indeterminate(res);
return;
}
best_K = 1;
best_log2err = 1e300;
/* todo: also break if too slow rate of decay? */
K_limit = 10 + prec * 0.25;
K_limit += pow(t, 0.2); /* possibly useful for off-strip evaluation */
for (K = 1; K < K_limit; K++)
{
if (sigma < 0 && K + sigma < 3)
continue;
/* Asymptotic approximation of the error term */
log2err = 2.7889996532222537064 - 0.12022458674074695061 / K +
0.2419040680416126037 * K + 0.7213475204444817037 * K * log(K)
+ (-0.7213475204444817037 - 0.7213475204444817037 * K) * log(t);
if (sigma >= 0.0)
log2err += -2.8073549220576041074 + 1.5 * sigma;
if (log2err < best_log2err)
{
best_log2err = log2err;
best_K = K;
}
if (log2err < -prec)
break;
}
K = best_K;
}
mag_init(err);
acb_dirichlet_zeta_rs_bound(err, s, K);
if (!mag_is_finite(err))
{
acb_indeterminate(res);
mag_clear(err);
return;
}
arb_init(a);
arb_init(p);
arb_init(api2);
arb_init(api2pow);
acb_init(U);
acb_init(S);
acb_init(u);
acb_init(v);
fmpz_init(N);
dk = _arb_vec_init((3 * K) / 2 + 2);
Fp = _acb_vec_init(3 * K + 1);
pipow = _arb_vec_init((3 * K) / 2 + 2);
for (wp = 2 * prec; ; wp *= 2)
{
/* a = sqrt(t / (2pi)) */
arb_const_pi(a, wp);
arb_mul_2exp_si(a, a, 1);
arb_div(a, acb_imagref(s), a, wp);
arb_sqrt(a, a, wp);
/* N = floor(a) */
arb_floor(p, a, wp);
if (!arb_get_unique_fmpz(N, p))
{
if (wp > 4 * prec && wp > arb_rel_accuracy_bits(acb_imagref(s)))
{
acb_indeterminate(res);
goto cleanup;
}
continue;
}
/* p = 1 + 2(N-a) */
arb_sub_fmpz(p, a, N, wp);
arb_neg(p, p);
arb_mul_2exp_si(p, p, 1);
arb_add_ui(p, p, 1, wp);
acb_dirichlet_zeta_rs_f_coeffs(Fp, p, 3 * K + 1, wp);
if (acb_rel_accuracy_bits(Fp + 3 * K) >= prec)
break;
if (wp > 4 * prec && wp > arb_rel_accuracy_bits(acb_imagref(s)))
break;
}
if (!fmpz_fits_si(N))
{
acb_indeterminate(res);
goto cleanup;
}
wp = prec + 10 + 3 * fmpz_bits(N); /* xxx */
wp = FLINT_MAX(wp, prec + 10);
wp = wp + FLINT_BIT_COUNT(K);
acb_zero(S);
arb_const_pi(api2, wp);
_arb_vec_set_powers(pipow, api2, (3 * K) / 2 + 2, wp);
arb_mul(api2, api2, api2, wp);
arb_mul(api2, api2, a, wp);
arb_inv(api2, api2, wp);
arb_one(api2pow);
for (k = 0; k <= K; k++)
{
acb_dirichlet_zeta_rs_d_coeffs(dk, acb_realref(s), k, wp);
acb_zero(u);
for (j = 0; j <= (3 * k) / 2; j++)
{
/* (pi/(2i))^j d^(k)_j F^(3k-2j)(p) */
arb_mul(acb_realref(v), pipow + j, dk + j, wp);
arb_mul_2exp_si(acb_realref(v), acb_realref(v), -j);
arb_zero(acb_imagref(v));
if (j % 4 == 1)
acb_div_onei(v, v);
else if (j % 4 == 2)
acb_neg(v, v);
else if (j % 4 == 3)
acb_mul_onei(v, v);
acb_addmul(u, v, Fp + 3 * k - 2 * j, wp);
}
acb_addmul_arb(S, u, api2pow, wp);
arb_mul(api2pow, api2pow, api2, wp);
}
acb_add_error_mag(S, err);
/* U = exp(-i[(t/2) log(t/(2pi)) - t/2 - pi/8]) */
arb_log(acb_realref(u), a, wp);
arb_mul_2exp_si(acb_realref(u), acb_realref(u), 1);
arb_sub_ui(acb_realref(u), acb_realref(u), 1, wp);
arb_mul(acb_realref(u), acb_realref(u), acb_imagref(s), wp);
arb_mul_2exp_si(acb_realref(u), acb_realref(u), -1);
arb_const_pi(acb_imagref(u), wp);
arb_mul_2exp_si(acb_imagref(u), acb_imagref(u), -3);
arb_sub(acb_realref(u), acb_realref(u), acb_imagref(u), wp);
arb_neg(acb_realref(u), acb_realref(u));
arb_sin_cos(acb_imagref(U), acb_realref(U), acb_realref(u), wp);
/* S = (-1)^(N-1) * U * a^(-sigma) * S */
acb_mul(S, S, U, wp);
arb_neg(acb_realref(u), acb_realref(s));
arb_pow(acb_realref(u), a, acb_realref(u), wp);
acb_mul_arb(S, S, acb_realref(u), wp);
if (fmpz_is_even(N))
acb_neg(S, S);
if (_acb_vec_estimate_allocated_bytes(fmpz_get_ui(N) / 6, wp) < 4e9)
acb_dirichlet_powsum_sieved(u, s, fmpz_get_ui(N), 1, wp);
else
acb_dirichlet_powsum_smooth(u, s, fmpz_get_ui(N), 1, wp);
acb_add(S, S, u, wp);
acb_set(res, S); /* don't set_round here; the extra precision is useful */
cleanup:
_arb_vec_clear(dk, (3 * K) / 2 + 2);
_acb_vec_clear(Fp, 3 * K + 1);
_arb_vec_clear(pipow, (3 * K) / 2 + 2);
arb_clear(a);
arb_clear(p);
arb_clear(api2);
arb_clear(api2pow);
acb_clear(U);
acb_clear(S);
acb_clear(u);
acb_clear(v);
fmpz_clear(N);
mag_clear(err);
}
|
