1 /* mpfr_zeta_ui -- compute the Riemann Zeta function for integer argument.
3 Copyright 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
4 Contributed by the Arenaire and Cacao 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 2.1 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.LIB. If not, write to
20 the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
21 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, mp_rnd_t r)
33 mpfr_set_ui (z, 1, r);
34 mpfr_div_2ui (z, z, 1, r);
46 mp_prec_t p = MPFR_PREC(z);
47 unsigned long n, k, err, kbits;
52 if (m >= p) /* 2^(-m) < ulp(1) = 2^(1-p). This means that
53 2^(-m) <= 1/2*ulp(1). We have 3^(-m)+4^(-m)+... < 2^(-m)
54 i.e. zeta(m) < 1+2*2^(-m) for m >= 3 */
57 if (m == 2) /* necessarily p=2 */
58 return mpfr_set_ui_2exp (z, 13, -3, r);
59 else if (r == GMP_RNDZ || r == GMP_RNDD || (r == GMP_RNDN && m > p))
61 mpfr_set_ui (z, 1, r);
66 mpfr_set_ui (z, 1, r);
72 /* now treat also the case where zeta(m) - (1+1/2^m) < 1/2*ulp(1),
73 and the result is either 1+2^(-m) or 1+2^(-m)+2^(1-p). */
76 if (m >= p / 2) /* otherwise 4^(-m) > 2^(-p) */
78 /* the following is a lower bound for log(3)/log(2) */
79 mpfr_set_str_binary (y, "1.100101011100000000011010001110");
80 mpfr_mul_ui (y, y, m, GMP_RNDZ); /* lower bound for log2(3^m) */
81 if (mpfr_cmp_ui (y, p + 2) >= 0)
84 mpfr_set_ui (z, 1, GMP_RNDZ);
85 mpfr_div_2ui (z, z, m, GMP_RNDZ);
86 mpfr_add_ui (z, z, 1, GMP_RNDZ);
99 p += MPFR_INT_CEIL_LOG2(p); /* account of the n term in the error */
101 p += MPFR_INT_CEIL_LOG2(p) + 15; /* initial value */
103 MPFR_ZIV_INIT (loop, p);
106 /* 0.39321985067869744 = log(2)/log(3+sqrt(8)) */
107 n = 1 + (unsigned long) (0.39321985067869744 * (double) p);
110 mpfr_set_prec (y, p);
112 /* computation of the d[k] */
115 mpz_mul_2exp (t, t, 2 * n - 1); /* t[n] */
117 for (k = n; k > 0; k--)
119 count_leading_zeros (kbits, k);
120 kbits = BITS_PER_MP_LIMB - kbits;
121 /* if k^m is too large, use mpz_tdiv_q */
122 if (m * kbits > 2 * BITS_PER_MP_LIMB)
124 /* if we know in advance that k^m > d, then floor(d/k^m) will
125 be zero below, so there is no need to compute k^m */
126 kbits = (kbits - 1) * m + 1;
127 /* k^m has at least kbits bits */
128 if (kbits > mpz_sizeinbase (d, 2))
132 mpz_ui_pow_ui (q, k, m);
133 mpz_tdiv_q (q, d, q);
136 else /* use several mpz_tdiv_q_ui calls */
138 unsigned long km = k, mm = m - 1;
139 while (mm > 0 && km < ULONG_MAX / k)
144 mpz_tdiv_q_ui (q, d, km);
149 while (mm > 0 && km < ULONG_MAX / k)
154 mpz_tdiv_q_ui (q, q, km);
162 /* we have d[k] = sum(t[i], i=k+1..n)
163 with t[i] = n*(n+i-1)!*4^i/(n-i)!/(2i)!
164 t[k-1]/t[k] = k*(2k-1)/(n-k+1)/(n+k-1)/2 */
165 #if (BITS_PER_MP_LIMB == 32)
166 #define KMAX 46341 /* max k such that k*(2k-1) < 2^32 */
167 #elif (BITS_PER_MP_LIMB == 64)
168 #define KMAX 3037000500
172 mpz_mul_ui (t, t, k * (2 * k - 1));
176 mpz_mul_ui (t, t, k);
177 mpz_mul_ui (t, t, 2 * k - 1);
179 mpz_div_2exp (t, t, 1);
180 /* Warning: the test below assumes that an unsigned long
181 has no padding bits. */
182 if (n < 1UL << ((sizeof(unsigned long) * CHAR_BIT) / 2))
183 /* (n - k + 1) * (n + k - 1) < n^2 */
184 mpz_divexact_ui (t, t, (n - k + 1) * (n + k - 1));
187 mpz_divexact_ui (t, t, n - k + 1);
188 mpz_divexact_ui (t, t, n + k - 1);
193 /* multiply by 1/(1-2^(1-m)) = 1 + 2^(1-m) + 2^(2-m) + ... */
194 mpz_div_2exp (t, s, m - 1);
199 mpz_div_2exp (t, t, m - 1);
201 while (mpz_cmp_ui (t, 0) > 0);
204 mpz_mul_2exp (s, s, p);
205 mpz_tdiv_q (s, s, d);
206 mpfr_set_z (y, s, GMP_RNDN);
207 mpfr_div_2ui (y, y, p, GMP_RNDN);
209 err = MPFR_INT_CEIL_LOG2 (err);
211 if (MPFR_LIKELY(MPFR_CAN_ROUND (y, p - err, MPFR_PREC(z), r)))
214 MPFR_ZIV_NEXT (loop, p);
216 MPFR_ZIV_FREE (loop);
222 inex = mpfr_set (z, y, r);