1 /* mpfr_zeta_ui -- compute the Riemann Zeta function for integer argument.
3 Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc.
4 Contributed by the Arenaire and Caramel projects, INRIA.
6 This file is part of the GNU MPFR Library.
8 The GNU MPFR Library is free software; you can redistribute it and/or modify
9 it under the terms of the GNU Lesser General Public License as published by
10 the Free Software Foundation; either version 3 of the License, or (at your
11 option) any later version.
13 The GNU MPFR Library is distributed in the hope that it will be useful, but
14 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
15 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
16 License for more details.
18 You should have received a copy of the GNU Lesser General Public License
19 along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
20 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
21 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
23 #define MPFR_NEED_LONGLONG_H
24 #include "mpfr-impl.h"
27 mpfr_zeta_ui (mpfr_ptr z, unsigned long m, mpfr_rnd_t r)
33 mpfr_set_ui (z, 1, r);
34 mpfr_div_2ui (z, z, 1, r);
47 mpfr_prec_t p = MPFR_PREC(z);
48 unsigned long n, k, err, kbits;
54 r = MPFR_RNDU; /* since the result is always positive */
56 if (m >= p) /* 2^(-m) < ulp(1) = 2^(1-p). This means that
57 2^(-m) <= 1/2*ulp(1). We have 3^(-m)+4^(-m)+... < 2^(-m)
58 i.e. zeta(m) < 1+2*2^(-m) for m >= 3 */
61 if (m == 2) /* necessarily p=2 */
62 return mpfr_set_ui_2exp (z, 13, -3, r);
63 else if (r == MPFR_RNDZ || r == MPFR_RNDD || (r == MPFR_RNDN && m > p))
65 mpfr_set_ui (z, 1, r);
70 mpfr_set_ui (z, 1, r);
76 /* now treat also the case where zeta(m) - (1+1/2^m) < 1/2*ulp(1),
77 and the result is either 1+2^(-m) or 1+2^(-m)+2^(1-p). */
80 if (m >= p / 2) /* otherwise 4^(-m) > 2^(-p) */
82 /* the following is a lower bound for log(3)/log(2) */
83 mpfr_set_str_binary (y, "1.100101011100000000011010001110");
84 mpfr_mul_ui (y, y, m, MPFR_RNDZ); /* lower bound for log2(3^m) */
85 if (mpfr_cmp_ui (y, p + 2) >= 0)
88 mpfr_set_ui (z, 1, MPFR_RNDZ);
89 mpfr_div_2ui (z, z, m, MPFR_RNDZ);
90 mpfr_add_ui (z, z, 1, MPFR_RNDZ);
103 p += MPFR_INT_CEIL_LOG2(p); /* account of the n term in the error */
105 p += MPFR_INT_CEIL_LOG2(p) + 15; /* initial value */
107 MPFR_ZIV_INIT (loop, p);
110 /* 0.39321985067869744 = log(2)/log(3+sqrt(8)) */
111 n = 1 + (unsigned long) (0.39321985067869744 * (double) p);
114 mpfr_set_prec (y, p);
116 /* computation of the d[k] */
119 mpz_mul_2exp (t, t, 2 * n - 1); /* t[n] */
121 for (k = n; k > 0; k--)
123 count_leading_zeros (kbits, k);
124 kbits = GMP_NUMB_BITS - kbits;
125 /* if k^m is too large, use mpz_tdiv_q */
126 if (m * kbits > 2 * GMP_NUMB_BITS)
128 /* if we know in advance that k^m > d, then floor(d/k^m) will
129 be zero below, so there is no need to compute k^m */
130 kbits = (kbits - 1) * m + 1;
131 /* k^m has at least kbits bits */
132 if (kbits > mpz_sizeinbase (d, 2))
136 mpz_ui_pow_ui (q, k, m);
137 mpz_tdiv_q (q, d, q);
140 else /* use several mpz_tdiv_q_ui calls */
142 unsigned long km = k, mm = m - 1;
143 while (mm > 0 && km < ULONG_MAX / k)
148 mpz_tdiv_q_ui (q, d, km);
153 while (mm > 0 && km < ULONG_MAX / k)
158 mpz_tdiv_q_ui (q, q, km);
166 /* we have d[k] = sum(t[i], i=k+1..n)
167 with t[i] = n*(n+i-1)!*4^i/(n-i)!/(2i)!
168 t[k-1]/t[k] = k*(2k-1)/(n-k+1)/(n+k-1)/2 */
169 #if (GMP_NUMB_BITS == 32)
170 #define KMAX 46341 /* max k such that k*(2k-1) < 2^32 */
171 #elif (GMP_NUMB_BITS == 64)
172 #define KMAX 3037000500
176 mpz_mul_ui (t, t, k * (2 * k - 1));
180 mpz_mul_ui (t, t, k);
181 mpz_mul_ui (t, t, 2 * k - 1);
183 mpz_fdiv_q_2exp (t, t, 1);
184 /* Warning: the test below assumes that an unsigned long
185 has no padding bits. */
186 if (n < 1UL << ((sizeof(unsigned long) * CHAR_BIT) / 2))
187 /* (n - k + 1) * (n + k - 1) < n^2 */
188 mpz_divexact_ui (t, t, (n - k + 1) * (n + k - 1));
191 mpz_divexact_ui (t, t, n - k + 1);
192 mpz_divexact_ui (t, t, n + k - 1);
197 /* multiply by 1/(1-2^(1-m)) = 1 + 2^(1-m) + 2^(2-m) + ... */
198 mpz_fdiv_q_2exp (t, s, m - 1);
203 mpz_fdiv_q_2exp (t, t, m - 1);
205 while (mpz_cmp_ui (t, 0) > 0);
208 mpz_mul_2exp (s, s, p);
209 mpz_tdiv_q (s, s, d);
210 mpfr_set_z (y, s, MPFR_RNDN);
211 mpfr_div_2ui (y, y, p, MPFR_RNDN);
213 err = MPFR_INT_CEIL_LOG2 (err);
215 if (MPFR_LIKELY(MPFR_CAN_ROUND (y, p - err, MPFR_PREC(z), r)))
218 MPFR_ZIV_NEXT (loop, p);
220 MPFR_ZIV_FREE (loop);
226 inex = mpfr_set (z, y, r);