1 //  Copyright John Maddock 2007, 2014.
2 //  Use, modification and distribution are subject to the
3 //  Boost Software License, Version 1.0. (See accompanying file
4 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
5 
6 #ifndef BOOST_MATH_ZETA_HPP
7 #define BOOST_MATH_ZETA_HPP
8 
9 #ifdef _MSC_VER
10 #pragma once
11 #endif
12 
13 #include <boost/math/special_functions/math_fwd.hpp>
14 #include <boost/math/tools/precision.hpp>
15 #include <boost/math/tools/series.hpp>
16 #include <boost/math/tools/big_constant.hpp>
17 #include <boost/math/policies/error_handling.hpp>
18 #include <boost/math/special_functions/gamma.hpp>
19 #include <boost/math/special_functions/factorials.hpp>
20 #include <boost/math/special_functions/sin_pi.hpp>
21 
22 #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
23 //
24 // This is the only way we can avoid
25 // warning: non-standard suffix on floating constant [-Wpedantic]
26 // when building with -Wall -pedantic.  Neither __extension__
27 // nor #pragma diagnostic ignored work :(
28 //
29 #pragma GCC system_header
30 #endif
31 
32 namespace boost{ namespace math{ namespace detail{
33 
34 #if 0
35 //
36 // This code is commented out because we have a better more rapidly converging series
37 // now.  Retained for future reference and in case the new code causes any issues down the line....
38 //
39 
40 template <class T, class Policy>
41 struct zeta_series_cache_size
42 {
43    //
44    // Work how large to make our cache size when evaluating the series
45    // evaluation:  normally this is just large enough for the series
46    // to have converged, but for arbitrary precision types we need a
47    // really large cache to achieve reasonable precision in a reasonable
48    // time.  This is important when constructing rational approximations
49    // to zeta for example.
50    //
51    typedef typename boost::math::policies::precision<T,Policy>::type precision_type;
52    typedef typename mpl::if_<
53       mpl::less_equal<precision_type, std::integral_constant<int, 0> >,
54       std::integral_constant<int, 5000>,
55       typename mpl::if_<
56          mpl::less_equal<precision_type, std::integral_constant<int, 64> >,
57          std::integral_constant<int, 70>,
58          typename mpl::if_<
59             mpl::less_equal<precision_type, std::integral_constant<int, 113> >,
60             std::integral_constant<int, 100>,
61             std::integral_constant<int, 5000>
62          >::type
63       >::type
64    >::type type;
65 };
66 
67 template <class T, class Policy>
68 T zeta_series_imp(T s, T sc, const Policy&)
69 {
70    //
71    // Series evaluation from:
72    // Havil, J. Gamma: Exploring Euler's Constant.
73    // Princeton, NJ: Princeton University Press, 2003.
74    //
75    // See also http://mathworld.wolfram.com/RiemannZetaFunction.html
76    //
77    BOOST_MATH_STD_USING
78    T sum = 0;
79    T mult = 0.5;
80    T change;
81    typedef typename zeta_series_cache_size<T,Policy>::type cache_size;
82    T powers[cache_size::value] = { 0, };
83    unsigned n = 0;
84    do{
85       T binom = -static_cast<T>(n);
86       T nested_sum = 1;
87       if(n < sizeof(powers) / sizeof(powers[0]))
88          powers[n] = pow(static_cast<T>(n + 1), -s);
89       for(unsigned k = 1; k <= n; ++k)
90       {
91          T p;
92          if(k < sizeof(powers) / sizeof(powers[0]))
93          {
94             p = powers[k];
95             //p = pow(k + 1, -s);
96          }
97          else
98             p = pow(static_cast<T>(k + 1), -s);
99          nested_sum += binom * p;
100         binom *= (k - static_cast<T>(n)) / (k + 1);
101       }
102       change = mult * nested_sum;
103       sum += change;
104       mult /= 2;
105       ++n;
106    }while(fabs(change / sum) > tools::epsilon<T>());
107 
108    return sum * 1 / -boost::math::powm1(T(2), sc);
109 }
110 
111 //
112 // Classical p-series:
113 //
114 template <class T>
115 struct zeta_series2
116 {
117    typedef T result_type;
118    zeta_series2(T _s) : s(-_s), k(1){}
119    T operator()()
120    {
121       BOOST_MATH_STD_USING
122       return pow(static_cast<T>(k++), s);
123    }
124 private:
125    T s;
126    unsigned k;
127 };
128 
129 template <class T, class Policy>
130 inline T zeta_series2_imp(T s, const Policy& pol)
131 {
132    std::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();;
133    zeta_series2<T> f(s);
134    T result = tools::sum_series(
135       f,
136       policies::get_epsilon<T, Policy>(),
137       max_iter);
138    policies::check_series_iterations<T>("boost::math::zeta_series2<%1%>(%1%)", max_iter, pol);
139    return result;
140 }
141 #endif
142 
143 template <class T, class Policy>
144 T zeta_polynomial_series(T s, T sc, Policy const &)
145 {
146    //
147    // This is algorithm 3 from:
148    //
149    // "An Efficient Algorithm for the Riemann Zeta Function", P. Borwein,
150    // Canadian Mathematical Society, Conference Proceedings.
151    // See: http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf
152    //
153    BOOST_MATH_STD_USING
154    int n = itrunc(T(log(boost::math::tools::epsilon<T>()) / -2));
155    T sum = 0;
156    T two_n = ldexp(T(1), n);
157    int ej_sign = 1;
158    for(int j = 0; j < n; ++j)
159    {
160       sum += ej_sign * -two_n / pow(T(j + 1), s);
161       ej_sign = -ej_sign;
162    }
163    T ej_sum = 1;
164    T ej_term = 1;
165    for(int j = n; j <= 2 * n - 1; ++j)
166    {
167       sum += ej_sign * (ej_sum - two_n) / pow(T(j + 1), s);
168       ej_sign = -ej_sign;
169       ej_term *= 2 * n - j;
170       ej_term /= j - n + 1;
171       ej_sum += ej_term;
172    }
173    return -sum / (two_n * (-powm1(T(2), sc)));
174 }
175 
176 template <class T, class Policy>
177 T zeta_imp_prec(T s, T sc, const Policy& pol, const std::integral_constant<int, 0>&)
178 {
179    BOOST_MATH_STD_USING
180    T result;
181    if(s >= policies::digits<T, Policy>())
182       return 1;
183    result = zeta_polynomial_series(s, sc, pol);
184 #if 0
185    // Old code archived for future reference:
186 
187    //
188    // Only use power series if it will converge in 100
189    // iterations or less: the more iterations it consumes
190    // the slower convergence becomes so we have to be very
191    // careful in it's usage.
192    //
193    if (s > -log(tools::epsilon<T>()) / 4.5)
194       result = detail::zeta_series2_imp(s, pol);
195    else
196       result = detail::zeta_series_imp(s, sc, pol);
197 #endif
198    return result;
199 }
200 
201 template <class T, class Policy>
zeta_imp_prec(T s,T sc,const Policy &,const std::integral_constant<int,53> &)202 inline T zeta_imp_prec(T s, T sc, const Policy&, const std::integral_constant<int, 53>&)
203 {
204    BOOST_MATH_STD_USING
205    T result;
206    if(s < 1)
207    {
208       // Rational Approximation
209       // Maximum Deviation Found:                     2.020e-18
210       // Expected Error Term:                         -2.020e-18
211       // Max error found at double precision:         3.994987e-17
212       static const T P[6] = {
213          static_cast<T>(0.24339294433593750202L),
214          static_cast<T>(-0.49092470516353571651L),
215          static_cast<T>(0.0557616214776046784287L),
216          static_cast<T>(-0.00320912498879085894856L),
217          static_cast<T>(0.000451534528645796438704L),
218          static_cast<T>(-0.933241270357061460782e-5L),
219         };
220       static const T Q[6] = {
221          static_cast<T>(1L),
222          static_cast<T>(-0.279960334310344432495L),
223          static_cast<T>(0.0419676223309986037706L),
224          static_cast<T>(-0.00413421406552171059003L),
225          static_cast<T>(0.00024978985622317935355L),
226          static_cast<T>(-0.101855788418564031874e-4L),
227       };
228       result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc);
229       result -= 1.2433929443359375F;
230       result += (sc);
231       result /= (sc);
232    }
233    else if(s <= 2)
234    {
235       // Maximum Deviation Found:        9.007e-20
236       // Expected Error Term:            9.007e-20
237       static const T P[6] = {
238          static_cast<T>(0.577215664901532860516L),
239          static_cast<T>(0.243210646940107164097L),
240          static_cast<T>(0.0417364673988216497593L),
241          static_cast<T>(0.00390252087072843288378L),
242          static_cast<T>(0.000249606367151877175456L),
243          static_cast<T>(0.110108440976732897969e-4L),
244       };
245       static const T Q[6] = {
246          static_cast<T>(1.0),
247          static_cast<T>(0.295201277126631761737L),
248          static_cast<T>(0.043460910607305495864L),
249          static_cast<T>(0.00434930582085826330659L),
250          static_cast<T>(0.000255784226140488490982L),
251          static_cast<T>(0.10991819782396112081e-4L),
252       };
253       result = tools::evaluate_polynomial(P, T(-sc)) / tools::evaluate_polynomial(Q, T(-sc));
254       result += 1 / (-sc);
255    }
256    else if(s <= 4)
257    {
258       // Maximum Deviation Found:          5.946e-22
259       // Expected Error Term:              -5.946e-22
260       static const float Y = 0.6986598968505859375;
261       static const T P[6] = {
262          static_cast<T>(-0.0537258300023595030676L),
263          static_cast<T>(0.0445163473292365591906L),
264          static_cast<T>(0.0128677673534519952905L),
265          static_cast<T>(0.00097541770457391752726L),
266          static_cast<T>(0.769875101573654070925e-4L),
267          static_cast<T>(0.328032510000383084155e-5L),
268       };
269       static const T Q[7] = {
270          1.0f,
271          static_cast<T>(0.33383194553034051422L),
272          static_cast<T>(0.0487798431291407621462L),
273          static_cast<T>(0.00479039708573558490716L),
274          static_cast<T>(0.000270776703956336357707L),
275          static_cast<T>(0.106951867532057341359e-4L),
276          static_cast<T>(0.236276623974978646399e-7L),
277       };
278       result = tools::evaluate_polynomial(P, T(s - 2)) / tools::evaluate_polynomial(Q, T(s - 2));
279       result += Y + 1 / (-sc);
280    }
281    else if(s <= 7)
282    {
283       // Maximum Deviation Found:                     2.955e-17
284       // Expected Error Term:                         2.955e-17
285       // Max error found at double precision:         2.009135e-16
286 
287       static const T P[6] = {
288          static_cast<T>(-2.49710190602259410021L),
289          static_cast<T>(-2.60013301809475665334L),
290          static_cast<T>(-0.939260435377109939261L),
291          static_cast<T>(-0.138448617995741530935L),
292          static_cast<T>(-0.00701721240549802377623L),
293          static_cast<T>(-0.229257310594893932383e-4L),
294       };
295       static const T Q[9] = {
296          1.0f,
297          static_cast<T>(0.706039025937745133628L),
298          static_cast<T>(0.15739599649558626358L),
299          static_cast<T>(0.0106117950976845084417L),
300          static_cast<T>(-0.36910273311764618902e-4L),
301          static_cast<T>(0.493409563927590008943e-5L),
302          static_cast<T>(-0.234055487025287216506e-6L),
303          static_cast<T>(0.718833729365459760664e-8L),
304          static_cast<T>(-0.1129200113474947419e-9L),
305       };
306       result = tools::evaluate_polynomial(P, T(s - 4)) / tools::evaluate_polynomial(Q, T(s - 4));
307       result = 1 + exp(result);
308    }
309    else if(s < 15)
310    {
311       // Maximum Deviation Found:                     7.117e-16
312       // Expected Error Term:                         7.117e-16
313       // Max error found at double precision:         9.387771e-16
314       static const T P[7] = {
315          static_cast<T>(-4.78558028495135619286L),
316          static_cast<T>(-1.89197364881972536382L),
317          static_cast<T>(-0.211407134874412820099L),
318          static_cast<T>(-0.000189204758260076688518L),
319          static_cast<T>(0.00115140923889178742086L),
320          static_cast<T>(0.639949204213164496988e-4L),
321          static_cast<T>(0.139348932445324888343e-5L),
322         };
323       static const T Q[9] = {
324          1.0f,
325          static_cast<T>(0.244345337378188557777L),
326          static_cast<T>(0.00873370754492288653669L),
327          static_cast<T>(-0.00117592765334434471562L),
328          static_cast<T>(-0.743743682899933180415e-4L),
329          static_cast<T>(-0.21750464515767984778e-5L),
330          static_cast<T>(0.471001264003076486547e-8L),
331          static_cast<T>(-0.833378440625385520576e-10L),
332          static_cast<T>(0.699841545204845636531e-12L),
333         };
334       result = tools::evaluate_polynomial(P, T(s - 7)) / tools::evaluate_polynomial(Q, T(s - 7));
335       result = 1 + exp(result);
336    }
337    else if(s < 36)
338    {
339       // Max error in interpolated form:             1.668e-17
340       // Max error found at long double precision:   1.669714e-17
341       static const T P[8] = {
342          static_cast<T>(-10.3948950573308896825L),
343          static_cast<T>(-2.85827219671106697179L),
344          static_cast<T>(-0.347728266539245787271L),
345          static_cast<T>(-0.0251156064655346341766L),
346          static_cast<T>(-0.00119459173416968685689L),
347          static_cast<T>(-0.382529323507967522614e-4L),
348          static_cast<T>(-0.785523633796723466968e-6L),
349          static_cast<T>(-0.821465709095465524192e-8L),
350       };
351       static const T Q[10] = {
352          1.0f,
353          static_cast<T>(0.208196333572671890965L),
354          static_cast<T>(0.0195687657317205033485L),
355          static_cast<T>(0.00111079638102485921877L),
356          static_cast<T>(0.408507746266039256231e-4L),
357          static_cast<T>(0.955561123065693483991e-6L),
358          static_cast<T>(0.118507153474022900583e-7L),
359          static_cast<T>(0.222609483627352615142e-14L),
360       };
361       result = tools::evaluate_polynomial(P, T(s - 15)) / tools::evaluate_polynomial(Q, T(s - 15));
362       result = 1 + exp(result);
363    }
364    else if(s < 56)
365    {
366       result = 1 + pow(T(2), -s);
367    }
368    else
369    {
370       result = 1;
371    }
372    return result;
373 }
374 
375 template <class T, class Policy>
376 T zeta_imp_prec(T s, T sc, const Policy&, const std::integral_constant<int, 64>&)
377 {
378    BOOST_MATH_STD_USING
379    T result;
380    if(s < 1)
381    {
382       // Rational Approximation
383       // Maximum Deviation Found:                     3.099e-20
384       // Expected Error Term:                         3.099e-20
385       // Max error found at long double precision:    5.890498e-20
386       static const T P[6] = {
387          BOOST_MATH_BIG_CONSTANT(T, 64, 0.243392944335937499969),
388          BOOST_MATH_BIG_CONSTANT(T, 64, -0.496837806864865688082),
389          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0680008039723709987107),
390          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00511620413006619942112),
391          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000455369899250053003335),
392          BOOST_MATH_BIG_CONSTANT(T, 64, -0.279496685273033761927e-4),
393         };
394       static const T Q[7] = {
395          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
396          BOOST_MATH_BIG_CONSTANT(T, 64, -0.30425480068225790522),
397          BOOST_MATH_BIG_CONSTANT(T, 64, 0.050052748580371598736),
398          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00519355671064700627862),
399          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000360623385771198350257),
400          BOOST_MATH_BIG_CONSTANT(T, 64, -0.159600883054550987633e-4),
401          BOOST_MATH_BIG_CONSTANT(T, 64, 0.339770279812410586032e-6),
402       };
403       result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc);
404       result -= 1.2433929443359375F;
405       result += (sc);
406       result /= (sc);
407    }
408    else if(s <= 2)
409    {
410       // Maximum Deviation Found:                     1.059e-21
411       // Expected Error Term:                         1.059e-21
412       // Max error found at long double precision:    1.626303e-19
413 
414       static const T P[6] = {
415          BOOST_MATH_BIG_CONSTANT(T, 64, 0.577215664901532860605),
416          BOOST_MATH_BIG_CONSTANT(T, 64, 0.222537368917162139445),
417          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0356286324033215682729),
418          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00304465292366350081446),
419          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000178102511649069421904),
420          BOOST_MATH_BIG_CONSTANT(T, 64, 0.700867470265983665042e-5),
421       };
422       static const T Q[7] = {
423          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
424          BOOST_MATH_BIG_CONSTANT(T, 64, 0.259385759149531030085),
425          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0373974962106091316854),
426          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00332735159183332820617),
427          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000188690420706998606469),
428          BOOST_MATH_BIG_CONSTANT(T, 64, 0.635994377921861930071e-5),
429          BOOST_MATH_BIG_CONSTANT(T, 64, 0.226583954978371199405e-7),
430       };
431       result = tools::evaluate_polynomial(P, T(-sc)) / tools::evaluate_polynomial(Q, T(-sc));
432       result += 1 / (-sc);
433    }
434    else if(s <= 4)
435    {
436       // Maximum Deviation Found:          5.946e-22
437       // Expected Error Term:              -5.946e-22
438       static const float Y = 0.6986598968505859375;
439       static const T P[7] = {
440          BOOST_MATH_BIG_CONSTANT(T, 64, -0.053725830002359501027),
441          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0470551187571475844778),
442          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0101339410415759517471),
443          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00100240326666092854528),
444          BOOST_MATH_BIG_CONSTANT(T, 64, 0.685027119098122814867e-4),
445          BOOST_MATH_BIG_CONSTANT(T, 64, 0.390972820219765942117e-5),
446          BOOST_MATH_BIG_CONSTANT(T, 64, 0.540319769113543934483e-7),
447       };
448       static const T Q[8] = {
449          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
450          BOOST_MATH_BIG_CONSTANT(T, 64, 0.286577739726542730421),
451          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0447355811517733225843),
452          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00430125107610252363302),
453          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000284956969089786662045),
454          BOOST_MATH_BIG_CONSTANT(T, 64, 0.116188101609848411329e-4),
455          BOOST_MATH_BIG_CONSTANT(T, 64, 0.278090318191657278204e-6),
456          BOOST_MATH_BIG_CONSTANT(T, 64, -0.19683620233222028478e-8),
457       };
458       result = tools::evaluate_polynomial(P, T(s - 2)) / tools::evaluate_polynomial(Q, T(s - 2));
459       result += Y + 1 / (-sc);
460    }
461    else if(s <= 7)
462    {
463       // Max error found at long double precision: 8.132216e-19
464       static const T P[8] = {
465          BOOST_MATH_BIG_CONSTANT(T, 64, -2.49710190602259407065),
466          BOOST_MATH_BIG_CONSTANT(T, 64, -3.36664913245960625334),
467          BOOST_MATH_BIG_CONSTANT(T, 64, -1.77180020623777595452),
468          BOOST_MATH_BIG_CONSTANT(T, 64, -0.464717885249654313933),
469          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0643694921293579472583),
470          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00464265386202805715487),
471          BOOST_MATH_BIG_CONSTANT(T, 64, -0.000165556579779704340166),
472          BOOST_MATH_BIG_CONSTANT(T, 64, -0.252884970740994069582e-5),
473       };
474       static const T Q[9] = {
475          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
476          BOOST_MATH_BIG_CONSTANT(T, 64, 1.01300131390690459085),
477          BOOST_MATH_BIG_CONSTANT(T, 64, 0.387898115758643503827),
478          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0695071490045701135188),
479          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00586908595251442839291),
480          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000217752974064612188616),
481          BOOST_MATH_BIG_CONSTANT(T, 64, 0.397626583349419011731e-5),
482          BOOST_MATH_BIG_CONSTANT(T, 64, -0.927884739284359700764e-8),
483          BOOST_MATH_BIG_CONSTANT(T, 64, 0.119810501805618894381e-9),
484       };
485       result = tools::evaluate_polynomial(P, T(s - 4)) / tools::evaluate_polynomial(Q, T(s - 4));
486       result = 1 + exp(result);
487    }
488    else if(s < 15)
489    {
490       // Max error in interpolated form:              1.133e-18
491       // Max error found at long double precision:    2.183198e-18
492       static const T P[9] = {
493          BOOST_MATH_BIG_CONSTANT(T, 64, -4.78558028495135548083),
494          BOOST_MATH_BIG_CONSTANT(T, 64, -3.23873322238609358947),
495          BOOST_MATH_BIG_CONSTANT(T, 64, -0.892338582881021799922),
496          BOOST_MATH_BIG_CONSTANT(T, 64, -0.131326296217965913809),
497          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0115651591773783712996),
498          BOOST_MATH_BIG_CONSTANT(T, 64, -0.000657728968362695775205),
499          BOOST_MATH_BIG_CONSTANT(T, 64, -0.252051328129449973047e-4),
500          BOOST_MATH_BIG_CONSTANT(T, 64, -0.626503445372641798925e-6),
501          BOOST_MATH_BIG_CONSTANT(T, 64, -0.815696314790853893484e-8),
502         };
503       static const T Q[9] = {
504          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
505          BOOST_MATH_BIG_CONSTANT(T, 64, 0.525765665400123515036),
506          BOOST_MATH_BIG_CONSTANT(T, 64, 0.10852641753657122787),
507          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0115669945375362045249),
508          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000732896513858274091966),
509          BOOST_MATH_BIG_CONSTANT(T, 64, 0.30683952282420248448e-4),
510          BOOST_MATH_BIG_CONSTANT(T, 64, 0.819649214609633126119e-6),
511          BOOST_MATH_BIG_CONSTANT(T, 64, 0.117957556472335968146e-7),
512          BOOST_MATH_BIG_CONSTANT(T, 64, -0.193432300973017671137e-12),
513         };
514       result = tools::evaluate_polynomial(P, T(s - 7)) / tools::evaluate_polynomial(Q, T(s - 7));
515       result = 1 + exp(result);
516    }
517    else if(s < 42)
518    {
519       // Max error in interpolated form:             1.668e-17
520       // Max error found at long double precision:   1.669714e-17
521       static const T P[9] = {
522          BOOST_MATH_BIG_CONSTANT(T, 64, -10.3948950573308861781),
523          BOOST_MATH_BIG_CONSTANT(T, 64, -2.82646012777913950108),
524          BOOST_MATH_BIG_CONSTANT(T, 64, -0.342144362739570333665),
525          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0249285145498722647472),
526          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00122493108848097114118),
527          BOOST_MATH_BIG_CONSTANT(T, 64, -0.423055371192592850196e-4),
528          BOOST_MATH_BIG_CONSTANT(T, 64, -0.1025215577185967488e-5),
529          BOOST_MATH_BIG_CONSTANT(T, 64, -0.165096762663509467061e-7),
530          BOOST_MATH_BIG_CONSTANT(T, 64, -0.145392555873022044329e-9),
531       };
532       static const T Q[10] = {
533          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
534          BOOST_MATH_BIG_CONSTANT(T, 64, 0.205135978585281988052),
535          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0192359357875879453602),
536          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00111496452029715514119),
537          BOOST_MATH_BIG_CONSTANT(T, 64, 0.434928449016693986857e-4),
538          BOOST_MATH_BIG_CONSTANT(T, 64, 0.116911068726610725891e-5),
539          BOOST_MATH_BIG_CONSTANT(T, 64, 0.206704342290235237475e-7),
540          BOOST_MATH_BIG_CONSTANT(T, 64, 0.209772836100827647474e-9),
541          BOOST_MATH_BIG_CONSTANT(T, 64, -0.939798249922234703384e-16),
542          BOOST_MATH_BIG_CONSTANT(T, 64, 0.264584017421245080294e-18),
543       };
544       result = tools::evaluate_polynomial(P, T(s - 15)) / tools::evaluate_polynomial(Q, T(s - 15));
545       result = 1 + exp(result);
546    }
547    else if(s < 63)
548    {
549       result = 1 + pow(T(2), -s);
550    }
551    else
552    {
553       result = 1;
554    }
555    return result;
556 }
557 
558 template <class T, class Policy>
559 T zeta_imp_prec(T s, T sc, const Policy&, const std::integral_constant<int, 113>&)
560 {
561    BOOST_MATH_STD_USING
562    T result;
563    if(s < 1)
564    {
565       // Rational Approximation
566       // Maximum Deviation Found:                     9.493e-37
567       // Expected Error Term:                         9.492e-37
568       // Max error found at long double precision:    7.281332e-31
569 
570       static const T P[10] = {
571          BOOST_MATH_BIG_CONSTANT(T, 113, -1.0),
572          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0353008629988648122808504280990313668),
573          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0107795651204927743049369868548706909),
574          BOOST_MATH_BIG_CONSTANT(T, 113, 0.000523961870530500751114866884685172975),
575          BOOST_MATH_BIG_CONSTANT(T, 113, -0.661805838304910731947595897966487515e-4),
576          BOOST_MATH_BIG_CONSTANT(T, 113, -0.658932670403818558510656304189164638e-5),
577          BOOST_MATH_BIG_CONSTANT(T, 113, -0.103437265642266106533814021041010453e-6),
578          BOOST_MATH_BIG_CONSTANT(T, 113, 0.116818787212666457105375746642927737e-7),
579          BOOST_MATH_BIG_CONSTANT(T, 113, 0.660690993901506912123512551294239036e-9),
580          BOOST_MATH_BIG_CONSTANT(T, 113, 0.113103113698388531428914333768142527e-10),
581         };
582       static const T Q[11] = {
583          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
584          BOOST_MATH_BIG_CONSTANT(T, 113, -0.387483472099602327112637481818565459),
585          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0802265315091063135271497708694776875),
586          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0110727276164171919280036408995078164),
587          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00112552716946286252000434849173787243),
588          BOOST_MATH_BIG_CONSTANT(T, 113, -0.874554160748626916455655180296834352e-4),
589          BOOST_MATH_BIG_CONSTANT(T, 113, 0.530097847491828379568636739662278322e-5),
590          BOOST_MATH_BIG_CONSTANT(T, 113, -0.248461553590496154705565904497247452e-6),
591          BOOST_MATH_BIG_CONSTANT(T, 113, 0.881834921354014787309644951507523899e-8),
592          BOOST_MATH_BIG_CONSTANT(T, 113, -0.217062446168217797598596496310953025e-9),
593          BOOST_MATH_BIG_CONSTANT(T, 113, 0.315823200002384492377987848307151168e-11),
594       };
595       result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc);
596       result += (sc);
597       result /= (sc);
598    }
599    else if(s <= 2)
600    {
601       // Maximum Deviation Found:                     1.616e-37
602       // Expected Error Term:                         -1.615e-37
603 
604       static const T P[10] = {
605          BOOST_MATH_BIG_CONSTANT(T, 113, 0.577215664901532860606512090082402431),
606          BOOST_MATH_BIG_CONSTANT(T, 113, 0.255597968739771510415479842335906308),
607          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0494056503552807274142218876983542205),
608          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00551372778611700965268920983472292325),
609          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00043667616723970574871427830895192731),
610          BOOST_MATH_BIG_CONSTANT(T, 113, 0.268562259154821957743669387915239528e-4),
611          BOOST_MATH_BIG_CONSTANT(T, 113, 0.109249633923016310141743084480436612e-5),
612          BOOST_MATH_BIG_CONSTANT(T, 113, 0.273895554345300227466534378753023924e-7),
613          BOOST_MATH_BIG_CONSTANT(T, 113, 0.583103205551702720149237384027795038e-9),
614          BOOST_MATH_BIG_CONSTANT(T, 113, -0.835774625259919268768735944711219256e-11),
615       };
616       static const T Q[11] = {
617          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
618          BOOST_MATH_BIG_CONSTANT(T, 113, 0.316661751179735502065583176348292881),
619          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0540401806533507064453851182728635272),
620          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00598621274107420237785899476374043797),
621          BOOST_MATH_BIG_CONSTANT(T, 113, 0.000474907812321704156213038740142079615),
622          BOOST_MATH_BIG_CONSTANT(T, 113, 0.272125421722314389581695715835862418e-4),
623          BOOST_MATH_BIG_CONSTANT(T, 113, 0.112649552156479800925522445229212933e-5),
624          BOOST_MATH_BIG_CONSTANT(T, 113, 0.301838975502992622733000078063330461e-7),
625          BOOST_MATH_BIG_CONSTANT(T, 113, 0.422960728687211282539769943184270106e-9),
626          BOOST_MATH_BIG_CONSTANT(T, 113, -0.377105263588822468076813329270698909e-11),
627          BOOST_MATH_BIG_CONSTANT(T, 113, -0.581926559304525152432462127383600681e-13),
628       };
629       result = tools::evaluate_polynomial(P, T(-sc)) / tools::evaluate_polynomial(Q, T(-sc));
630       result += 1 / (-sc);
631    }
632    else if(s <= 4)
633    {
634       // Maximum Deviation Found:                     1.891e-36
635       // Expected Error Term:                         -1.891e-36
636       // Max error found: 2.171527e-35
637 
638       static const float Y = 0.6986598968505859375;
639       static const T P[11] = {
640          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0537258300023595010275848333539748089),
641          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0429086930802630159457448174466342553),
642          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0136148228754303412510213395034056857),
643          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00190231601036042925183751238033763915),
644          BOOST_MATH_BIG_CONSTANT(T, 113, 0.000186880390916311438818302549192456581),
645          BOOST_MATH_BIG_CONSTANT(T, 113, 0.145347370745893262394287982691323657e-4),
646          BOOST_MATH_BIG_CONSTANT(T, 113, 0.805843276446813106414036600485884885e-6),
647          BOOST_MATH_BIG_CONSTANT(T, 113, 0.340818159286739137503297172091882574e-7),
648          BOOST_MATH_BIG_CONSTANT(T, 113, 0.115762357488748996526167305116837246e-8),
649          BOOST_MATH_BIG_CONSTANT(T, 113, 0.231904754577648077579913403645767214e-10),
650          BOOST_MATH_BIG_CONSTANT(T, 113, 0.340169592866058506675897646629036044e-12),
651       };
652       static const T Q[12] = {
653          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
654          BOOST_MATH_BIG_CONSTANT(T, 113, 0.363755247765087100018556983050520554),
655          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0696581979014242539385695131258321598),
656          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00882208914484611029571547753782014817),
657          BOOST_MATH_BIG_CONSTANT(T, 113, 0.000815405623261946661762236085660996718),
658          BOOST_MATH_BIG_CONSTANT(T, 113, 0.571366167062457197282642344940445452e-4),
659          BOOST_MATH_BIG_CONSTANT(T, 113, 0.309278269271853502353954062051797838e-5),
660          BOOST_MATH_BIG_CONSTANT(T, 113, 0.12822982083479010834070516053794262e-6),
661          BOOST_MATH_BIG_CONSTANT(T, 113, 0.397876357325018976733953479182110033e-8),
662          BOOST_MATH_BIG_CONSTANT(T, 113, 0.8484432107648683277598472295289279e-10),
663          BOOST_MATH_BIG_CONSTANT(T, 113, 0.105677416606909614301995218444080615e-11),
664          BOOST_MATH_BIG_CONSTANT(T, 113, 0.547223964564003701979951154093005354e-15),
665       };
666       result = tools::evaluate_polynomial(P, T(s - 2)) / tools::evaluate_polynomial(Q, T(s - 2));
667       result += Y + 1 / (-sc);
668    }
669    else if(s <= 6)
670    {
671       // Max error in interpolated form:             1.510e-37
672       // Max error found at long double precision:   2.769266e-34
673 
674       static const T Y = 3.28348541259765625F;
675 
676       static const T P[13] = {
677          BOOST_MATH_BIG_CONSTANT(T, 113, 0.786383506575062179339611614117697622),
678          BOOST_MATH_BIG_CONSTANT(T, 113, 0.495766593395271370974685959652073976),
679          BOOST_MATH_BIG_CONSTANT(T, 113, -0.409116737851754766422360889037532228),
680          BOOST_MATH_BIG_CONSTANT(T, 113, -0.57340744006238263817895456842655987),
681          BOOST_MATH_BIG_CONSTANT(T, 113, -0.280479899797421910694892949057963111),
682          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0753148409447590257157585696212649869),
683          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0122934003684672788499099362823748632),
684          BOOST_MATH_BIG_CONSTANT(T, 113, -0.00126148398446193639247961370266962927),
685          BOOST_MATH_BIG_CONSTANT(T, 113, -0.828465038179772939844657040917364896e-4),
686          BOOST_MATH_BIG_CONSTANT(T, 113, -0.361008916706050977143208468690645684e-5),
687          BOOST_MATH_BIG_CONSTANT(T, 113, -0.109879825497910544424797771195928112e-6),
688          BOOST_MATH_BIG_CONSTANT(T, 113, -0.214539416789686920918063075528797059e-8),
689          BOOST_MATH_BIG_CONSTANT(T, 113, -0.15090220092460596872172844424267351e-10),
690       };
691       static const T Q[14] = {
692          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
693          BOOST_MATH_BIG_CONSTANT(T, 113, 1.69490865837142338462982225731926485),
694          BOOST_MATH_BIG_CONSTANT(T, 113, 1.22697696630994080733321401255942464),
695          BOOST_MATH_BIG_CONSTANT(T, 113, 0.495409420862526540074366618006341533),
696          BOOST_MATH_BIG_CONSTANT(T, 113, 0.122368084916843823462872905024259633),
697          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0191412993625268971656513890888208623),
698          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00191401538628980617753082598351559642),
699          BOOST_MATH_BIG_CONSTANT(T, 113, 0.000123318142456272424148930280876444459),
700          BOOST_MATH_BIG_CONSTANT(T, 113, 0.531945488232526067889835342277595709e-5),
701          BOOST_MATH_BIG_CONSTANT(T, 113, 0.161843184071894368337068779669116236e-6),
702          BOOST_MATH_BIG_CONSTANT(T, 113, 0.305796079600152506743828859577462778e-8),
703          BOOST_MATH_BIG_CONSTANT(T, 113, 0.233582592298450202680170811044408894e-10),
704          BOOST_MATH_BIG_CONSTANT(T, 113, -0.275363878344548055574209713637734269e-13),
705          BOOST_MATH_BIG_CONSTANT(T, 113, 0.221564186807357535475441900517843892e-15),
706       };
707       result = tools::evaluate_polynomial(P, T(s - 4)) / tools::evaluate_polynomial(Q, T(s - 4));
708       result -= Y;
709       result = 1 + exp(result);
710    }
711    else if(s < 10)
712    {
713       // Max error in interpolated form:             1.999e-34
714       // Max error found at long double precision:   2.156186e-33
715 
716       static const T P[13] = {
717          BOOST_MATH_BIG_CONSTANT(T, 113, -4.0545627381873738086704293881227365),
718          BOOST_MATH_BIG_CONSTANT(T, 113, -4.70088348734699134347906176097717782),
719          BOOST_MATH_BIG_CONSTANT(T, 113, -2.36921550900925512951976617607678789),
720          BOOST_MATH_BIG_CONSTANT(T, 113, -0.684322583796369508367726293719322866),
721          BOOST_MATH_BIG_CONSTANT(T, 113, -0.126026534540165129870721937592996324),
722          BOOST_MATH_BIG_CONSTANT(T, 113, -0.015636903921778316147260572008619549),
723          BOOST_MATH_BIG_CONSTANT(T, 113, -0.00135442294754728549644376325814460807),
724          BOOST_MATH_BIG_CONSTANT(T, 113, -0.842793965853572134365031384646117061e-4),
725          BOOST_MATH_BIG_CONSTANT(T, 113, -0.385602133791111663372015460784978351e-5),
726          BOOST_MATH_BIG_CONSTANT(T, 113, -0.130458500394692067189883214401478539e-6),
727          BOOST_MATH_BIG_CONSTANT(T, 113, -0.315861074947230418778143153383660035e-8),
728          BOOST_MATH_BIG_CONSTANT(T, 113, -0.500334720512030826996373077844707164e-10),
729          BOOST_MATH_BIG_CONSTANT(T, 113, -0.420204769185233365849253969097184005e-12),
730         };
731       static const T Q[14] = {
732          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
733          BOOST_MATH_BIG_CONSTANT(T, 113, 0.97663511666410096104783358493318814),
734          BOOST_MATH_BIG_CONSTANT(T, 113, 0.40878780231201806504987368939673249),
735          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0963890666609396058945084107597727252),
736          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0142207619090854604824116070866614505),
737          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00139010220902667918476773423995750877),
738          BOOST_MATH_BIG_CONSTANT(T, 113, 0.940669540194694997889636696089994734e-4),
739          BOOST_MATH_BIG_CONSTANT(T, 113, 0.458220848507517004399292480807026602e-5),
740          BOOST_MATH_BIG_CONSTANT(T, 113, 0.16345521617741789012782420625435495e-6),
741          BOOST_MATH_BIG_CONSTANT(T, 113, 0.414007452533083304371566316901024114e-8),
742          BOOST_MATH_BIG_CONSTANT(T, 113, 0.68701473543366328016953742622661377e-10),
743          BOOST_MATH_BIG_CONSTANT(T, 113, 0.603461891080716585087883971886075863e-12),
744          BOOST_MATH_BIG_CONSTANT(T, 113, 0.294670713571839023181857795866134957e-16),
745          BOOST_MATH_BIG_CONSTANT(T, 113, -0.147003914536437243143096875069813451e-18),
746         };
747       result = tools::evaluate_polynomial(P, T(s - 6)) / tools::evaluate_polynomial(Q, T(s - 6));
748       result = 1 + exp(result);
749    }
750    else if(s < 17)
751    {
752       // Max error in interpolated form:             1.641e-32
753       // Max error found at long double precision:   1.696121e-32
754       static const T P[13] = {
755          BOOST_MATH_BIG_CONSTANT(T, 113, -6.91319491921722925920883787894829678),
756          BOOST_MATH_BIG_CONSTANT(T, 113, -3.65491257639481960248690596951049048),
757          BOOST_MATH_BIG_CONSTANT(T, 113, -0.813557553449954526442644544105257881),
758          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0994317301685870959473658713841138083),
759          BOOST_MATH_BIG_CONSTANT(T, 113, -0.00726896610245676520248617014211734906),
760          BOOST_MATH_BIG_CONSTANT(T, 113, -0.000317253318715075854811266230916762929),
761          BOOST_MATH_BIG_CONSTANT(T, 113, -0.66851422826636750855184211580127133e-5),
762          BOOST_MATH_BIG_CONSTANT(T, 113, 0.879464154730985406003332577806849971e-7),
763          BOOST_MATH_BIG_CONSTANT(T, 113, 0.113838903158254250631678791998294628e-7),
764          BOOST_MATH_BIG_CONSTANT(T, 113, 0.379184410304927316385211327537817583e-9),
765          BOOST_MATH_BIG_CONSTANT(T, 113, 0.612992858643904887150527613446403867e-11),
766          BOOST_MATH_BIG_CONSTANT(T, 113, 0.347873737198164757035457841688594788e-13),
767          BOOST_MATH_BIG_CONSTANT(T, 113, -0.289187187441625868404494665572279364e-15),
768         };
769       static const T Q[14] = {
770          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
771          BOOST_MATH_BIG_CONSTANT(T, 113, 0.427310044448071818775721584949868806),
772          BOOST_MATH_BIG_CONSTANT(T, 113, 0.074602514873055756201435421385243062),
773          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00688651562174480772901425121653945942),
774          BOOST_MATH_BIG_CONSTANT(T, 113, 0.000360174847635115036351323894321880445),
775          BOOST_MATH_BIG_CONSTANT(T, 113, 0.973556847713307543918865405758248777e-5),
776          BOOST_MATH_BIG_CONSTANT(T, 113, -0.853455848314516117964634714780874197e-8),
777          BOOST_MATH_BIG_CONSTANT(T, 113, -0.118203513654855112421673192194622826e-7),
778          BOOST_MATH_BIG_CONSTANT(T, 113, -0.462521662511754117095006543363328159e-9),
779          BOOST_MATH_BIG_CONSTANT(T, 113, -0.834212591919475633107355719369463143e-11),
780          BOOST_MATH_BIG_CONSTANT(T, 113, -0.5354594751002702935740220218582929e-13),
781          BOOST_MATH_BIG_CONSTANT(T, 113, 0.406451690742991192964889603000756203e-15),
782          BOOST_MATH_BIG_CONSTANT(T, 113, 0.887948682401000153828241615760146728e-19),
783          BOOST_MATH_BIG_CONSTANT(T, 113, -0.34980761098820347103967203948619072e-21),
784         };
785       result = tools::evaluate_polynomial(P, T(s - 10)) / tools::evaluate_polynomial(Q, T(s - 10));
786       result = 1 + exp(result);
787    }
788    else if(s < 30)
789    {
790       // Max error in interpolated form:             1.563e-31
791       // Max error found at long double precision:   1.562725e-31
792 
793       static const T P[13] = {
794          BOOST_MATH_BIG_CONSTANT(T, 113, -11.7824798233959252791987402769438322),
795          BOOST_MATH_BIG_CONSTANT(T, 113, -4.36131215284987731928174218354118102),
796          BOOST_MATH_BIG_CONSTANT(T, 113, -0.732260980060982349410898496846972204),
797          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0744985185694913074484248803015717388),
798          BOOST_MATH_BIG_CONSTANT(T, 113, -0.00517228281320594683022294996292250527),
799          BOOST_MATH_BIG_CONSTANT(T, 113, -0.000260897206152101522569969046299309939),
800          BOOST_MATH_BIG_CONSTANT(T, 113, -0.989553462123121764865178453128769948e-5),
801          BOOST_MATH_BIG_CONSTANT(T, 113, -0.286916799741891410827712096608826167e-6),
802          BOOST_MATH_BIG_CONSTANT(T, 113, -0.637262477796046963617949532211619729e-8),
803          BOOST_MATH_BIG_CONSTANT(T, 113, -0.106796831465628373325491288787760494e-9),
804          BOOST_MATH_BIG_CONSTANT(T, 113, -0.129343095511091870860498356205376823e-11),
805          BOOST_MATH_BIG_CONSTANT(T, 113, -0.102397936697965977221267881716672084e-13),
806          BOOST_MATH_BIG_CONSTANT(T, 113, -0.402663128248642002351627980255756363e-16),
807       };
808       static const T Q[14] = {
809          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
810          BOOST_MATH_BIG_CONSTANT(T, 113, 0.311288325355705609096155335186466508),
811          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0438318468940415543546769437752132748),
812          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00374396349183199548610264222242269536),
813          BOOST_MATH_BIG_CONSTANT(T, 113, 0.000218707451200585197339671707189281302),
814          BOOST_MATH_BIG_CONSTANT(T, 113, 0.927578767487930747532953583797351219e-5),
815          BOOST_MATH_BIG_CONSTANT(T, 113, 0.294145760625753561951137473484889639e-6),
816          BOOST_MATH_BIG_CONSTANT(T, 113, 0.704618586690874460082739479535985395e-8),
817          BOOST_MATH_BIG_CONSTANT(T, 113, 0.126333332872897336219649130062221257e-9),
818          BOOST_MATH_BIG_CONSTANT(T, 113, 0.16317315713773503718315435769352765e-11),
819          BOOST_MATH_BIG_CONSTANT(T, 113, 0.137846712823719515148344938160275695e-13),
820          BOOST_MATH_BIG_CONSTANT(T, 113, 0.580975420554224366450994232723910583e-16),
821          BOOST_MATH_BIG_CONSTANT(T, 113, -0.291354445847552426900293580511392459e-22),
822          BOOST_MATH_BIG_CONSTANT(T, 113, 0.73614324724785855925025452085443636e-25),
823       };
824       result = tools::evaluate_polynomial(P, T(s - 17)) / tools::evaluate_polynomial(Q, T(s - 17));
825       result = 1 + exp(result);
826    }
827    else if(s < 74)
828    {
829       // Max error in interpolated form:             2.311e-27
830       // Max error found at long double precision:   2.297544e-27
831       static const T P[14] = {
832          BOOST_MATH_BIG_CONSTANT(T, 113, -20.7944102007844314586649688802236072),
833          BOOST_MATH_BIG_CONSTANT(T, 113, -4.95759941987499442499908748130192187),
834          BOOST_MATH_BIG_CONSTANT(T, 113, -0.563290752832461751889194629200298688),
835          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0406197001137935911912457120706122877),
836          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0020846534789473022216888863613422293),
837          BOOST_MATH_BIG_CONSTANT(T, 113, -0.808095978462109173749395599401375667e-4),
838          BOOST_MATH_BIG_CONSTANT(T, 113, -0.244706022206249301640890603610060959e-5),
839          BOOST_MATH_BIG_CONSTANT(T, 113, -0.589477682919645930544382616501666572e-7),
840          BOOST_MATH_BIG_CONSTANT(T, 113, -0.113699573675553496343617442433027672e-8),
841          BOOST_MATH_BIG_CONSTANT(T, 113, -0.174767860183598149649901223128011828e-10),
842          BOOST_MATH_BIG_CONSTANT(T, 113, -0.210051620306761367764549971980026474e-12),
843          BOOST_MATH_BIG_CONSTANT(T, 113, -0.189187969537370950337212675466400599e-14),
844          BOOST_MATH_BIG_CONSTANT(T, 113, -0.116313253429564048145641663778121898e-16),
845          BOOST_MATH_BIG_CONSTANT(T, 113, -0.376708747782400769427057630528578187e-19),
846       };
847       static const T Q[16] = {
848          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
849          BOOST_MATH_BIG_CONSTANT(T, 113, 0.205076752981410805177554569784219717),
850          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0202526722696670378999575738524540269),
851          BOOST_MATH_BIG_CONSTANT(T, 113, 0.001278305290005994980069466658219057),
852          BOOST_MATH_BIG_CONSTANT(T, 113, 0.576404779858501791742255670403304787e-4),
853          BOOST_MATH_BIG_CONSTANT(T, 113, 0.196477049872253010859712483984252067e-5),
854          BOOST_MATH_BIG_CONSTANT(T, 113, 0.521863830500876189501054079974475762e-7),
855          BOOST_MATH_BIG_CONSTANT(T, 113, 0.109524209196868135198775445228552059e-8),
856          BOOST_MATH_BIG_CONSTANT(T, 113, 0.181698713448644481083966260949267825e-10),
857          BOOST_MATH_BIG_CONSTANT(T, 113, 0.234793316975091282090312036524695562e-12),
858          BOOST_MATH_BIG_CONSTANT(T, 113, 0.227490441461460571047545264251399048e-14),
859          BOOST_MATH_BIG_CONSTANT(T, 113, 0.151500292036937400913870642638520668e-16),
860          BOOST_MATH_BIG_CONSTANT(T, 113, 0.543475775154780935815530649335936121e-19),
861          BOOST_MATH_BIG_CONSTANT(T, 113, 0.241647013434111434636554455083309352e-28),
862          BOOST_MATH_BIG_CONSTANT(T, 113, -0.557103423021951053707162364713587374e-31),
863          BOOST_MATH_BIG_CONSTANT(T, 113, 0.618708773442584843384712258199645166e-34),
864       };
865       result = tools::evaluate_polynomial(P, T(s - 30)) / tools::evaluate_polynomial(Q, T(s - 30));
866       result = 1 + exp(result);
867    }
868    else if(s < 117)
869    {
870       result = 1 + pow(T(2), -s);
871    }
872    else
873    {
874       result = 1;
875    }
876    return result;
877 }
878 
879 template <class T, class Policy>
880 T zeta_imp_odd_integer(int s, const T&, const Policy&, const std::true_type&)
881 {
882    static const T results[] = {
883       BOOST_MATH_BIG_CONSTANT(T, 113, 1.2020569031595942853997381615114500), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0369277551433699263313654864570342), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0083492773819228268397975498497968), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0020083928260822144178527692324121), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0004941886041194645587022825264699), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0001227133475784891467518365263574), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000305882363070204935517285106451), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000076371976378997622736002935630), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000019082127165539389256569577951), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000004769329867878064631167196044), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000001192199259653110730677887189), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000298035035146522801860637051), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000074507117898354294919810042), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000018626597235130490064039099), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000004656629065033784072989233), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000001164155017270051977592974), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000291038504449709968692943), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000072759598350574810145209), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000018189896503070659475848), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000004547473783042154026799), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000001136868407680227849349), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000284217097688930185546), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000071054273952108527129), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000017763568435791203275), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000004440892103143813364), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000001110223025141066134), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000277555756213612417), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000069388939045441537), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000017347234760475766), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000004336808690020650), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000001084202172494241), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000271050543122347), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000067762635780452), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000016940658945098), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000004235164736273), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000001058791184068), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000264697796017), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000066174449004), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000016543612251), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000004135903063), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000001033975766), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000258493941), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000064623485), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000016155871), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000004038968), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000001009742), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000252435), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000063109), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000015777), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000003944), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000986), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000247), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000062), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000015), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000004), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000001),
884    };
885    return s > 113 ? 1 : results[(s - 3) / 2];
886 }
887 
888 template <class T, class Policy>
889 T zeta_imp_odd_integer(int s, const T& sc, const Policy& pol, const std::false_type&)
890 {
891 #ifdef BOOST_MATH_NO_THREAD_LOCAL_WITH_NON_TRIVIAL_TYPES
892    static_assert(std::is_trivially_destructible<T>::value, "Your platform does not support thread_local with non-trivial types, last checked with Mingw-x64-8.1, Jan 2021.  Please try a Mingw build with the POSIX threading model, see https://sourceforge.net/p/mingw-w64/bugs/527/");
893 #endif
894    static BOOST_MATH_THREAD_LOCAL bool is_init = false;
895    static BOOST_MATH_THREAD_LOCAL T results[50] = {};
896    static BOOST_MATH_THREAD_LOCAL int digits = tools::digits<T>();
897    int current_digits = tools::digits<T>();
898    if(digits != current_digits)
899    {
900       // Oh my precision has changed...
901       is_init = false;
902    }
903    if(!is_init)
904    {
905       is_init = true;
906       digits = current_digits;
907       for(unsigned k = 0; k < sizeof(results) / sizeof(results[0]); ++k)
908       {
909          T arg = k * 2 + 3;
910          T c_arg = 1 - arg;
911          results[k] = zeta_polynomial_series(arg, c_arg, pol);
912       }
913    }
914    unsigned index = (s - 3) / 2;
915    return index >= sizeof(results) / sizeof(results[0]) ? zeta_polynomial_series(T(s), sc, pol): results[index];
916 }
917 
918 template <class T, class Policy, class Tag>
919 T zeta_imp(T s, T sc, const Policy& pol, const Tag& tag)
920 {
921    BOOST_MATH_STD_USING
922    static const char* function = "boost::math::zeta<%1%>";
923    if(sc == 0)
924       return policies::raise_pole_error<T>(
925          function,
926          "Evaluation of zeta function at pole %1%",
927          s, pol);
928    T result;
929    //
930    // Trivial case:
931    //
932    if(s > policies::digits<T, Policy>())
933       return 1;
934    //
935    // Start by seeing if we have a simple closed form:
936    //
937    if(floor(s) == s)
938    {
939 #ifndef BOOST_NO_EXCEPTIONS
940       // Without exceptions we expect itrunc to return INT_MAX on overflow
941       // and we fall through anyway.
942       try
943       {
944 #endif
945          int v = itrunc(s);
946          if(v == s)
947          {
948             if(v < 0)
949             {
950                if(((-v) & 1) == 0)
951                   return 0;
952                int n = (-v + 1) / 2;
953                if(n <= (int)boost::math::max_bernoulli_b2n<T>::value)
954                   return T((-v & 1) ? -1 : 1) * boost::math::unchecked_bernoulli_b2n<T>(n) / (1 - v);
955             }
956             else if((v & 1) == 0)
957             {
958                if(((v / 2) <= (int)boost::math::max_bernoulli_b2n<T>::value) && (v <= (int)boost::math::max_factorial<T>::value))
959                   return T(((v / 2 - 1) & 1) ? -1 : 1) * ldexp(T(1), v - 1) * pow(constants::pi<T, Policy>(), v) *
960                      boost::math::unchecked_bernoulli_b2n<T>(v / 2) / boost::math::unchecked_factorial<T>(v);
961                return T(((v / 2 - 1) & 1) ? -1 : 1) * ldexp(T(1), v - 1) * pow(constants::pi<T, Policy>(), v) *
962                   boost::math::bernoulli_b2n<T>(v / 2) / boost::math::factorial<T>(v, pol);
963             }
964             else
965                return zeta_imp_odd_integer(v, sc, pol, std::integral_constant<bool, (Tag::value <= 113) && Tag::value>());
966          }
967 #ifndef BOOST_NO_EXCEPTIONS
968       }
969       catch(const boost::math::rounding_error&){} // Just fall through, s is too large to round
970       catch(const std::overflow_error&){}
971 #endif
972    }
973 
974    if(fabs(s) < tools::root_epsilon<T>())
975    {
976       result = -0.5f - constants::log_root_two_pi<T, Policy>() * s;
977    }
978    else if(s < 0)
979    {
980       std::swap(s, sc);
981       if(floor(sc/2) == sc/2)
982          result = 0;
983       else
984       {
985          if(s > max_factorial<T>::value)
986          {
987             T mult = boost::math::sin_pi(0.5f * sc, pol) * 2 * zeta_imp(s, sc, pol, tag);
988             result = boost::math::lgamma(s, pol);
989             result -= s * log(2 * constants::pi<T>());
990             if(result > tools::log_max_value<T>())
991                return sign(mult) * policies::raise_overflow_error<T>(function, 0, pol);
992             result = exp(result);
993             if(tools::max_value<T>() / fabs(mult) < result)
994                return boost::math::sign(mult) * policies::raise_overflow_error<T>(function, 0, pol);
995             result *= mult;
996          }
997          else
998          {
999             result = boost::math::sin_pi(0.5f * sc, pol)
1000                * 2 * pow(2 * constants::pi<T>(), -s)
1001                * boost::math::tgamma(s, pol)
1002                * zeta_imp(s, sc, pol, tag);
1003          }
1004       }
1005    }
1006    else
1007    {
1008       result = zeta_imp_prec(s, sc, pol, tag);
1009    }
1010    return result;
1011 }
1012 
1013 template <class T, class Policy, class tag>
1014 struct zeta_initializer
1015 {
1016    struct init
1017    {
initboost::math::detail::zeta_initializer::init1018       init()
1019       {
1020          do_init(tag());
1021       }
do_initboost::math::detail::zeta_initializer::init1022       static void do_init(const std::integral_constant<int, 0>&){ boost::math::zeta(static_cast<T>(5), Policy()); }
do_initboost::math::detail::zeta_initializer::init1023       static void do_init(const std::integral_constant<int, 53>&){ boost::math::zeta(static_cast<T>(5), Policy()); }
do_initboost::math::detail::zeta_initializer::init1024       static void do_init(const std::integral_constant<int, 64>&)
1025       {
1026          boost::math::zeta(static_cast<T>(0.5), Policy());
1027          boost::math::zeta(static_cast<T>(1.5), Policy());
1028          boost::math::zeta(static_cast<T>(3.5), Policy());
1029          boost::math::zeta(static_cast<T>(6.5), Policy());
1030          boost::math::zeta(static_cast<T>(14.5), Policy());
1031          boost::math::zeta(static_cast<T>(40.5), Policy());
1032 
1033          boost::math::zeta(static_cast<T>(5), Policy());
1034       }
do_initboost::math::detail::zeta_initializer::init1035       static void do_init(const std::integral_constant<int, 113>&)
1036       {
1037          boost::math::zeta(static_cast<T>(0.5), Policy());
1038          boost::math::zeta(static_cast<T>(1.5), Policy());
1039          boost::math::zeta(static_cast<T>(3.5), Policy());
1040          boost::math::zeta(static_cast<T>(5.5), Policy());
1041          boost::math::zeta(static_cast<T>(9.5), Policy());
1042          boost::math::zeta(static_cast<T>(16.5), Policy());
1043          boost::math::zeta(static_cast<T>(25.5), Policy());
1044          boost::math::zeta(static_cast<T>(70.5), Policy());
1045 
1046          boost::math::zeta(static_cast<T>(5), Policy());
1047       }
force_instantiateboost::math::detail::zeta_initializer::init1048       void force_instantiate()const{}
1049    };
1050    static const init initializer;
force_instantiateboost::math::detail::zeta_initializer1051    static void force_instantiate()
1052    {
1053       initializer.force_instantiate();
1054    }
1055 };
1056 
1057 template <class T, class Policy, class tag>
1058 const typename zeta_initializer<T, Policy, tag>::init zeta_initializer<T, Policy, tag>::initializer;
1059 
1060 } // detail
1061 
1062 template <class T, class Policy>
zeta(T s,const Policy &)1063 inline typename tools::promote_args<T>::type zeta(T s, const Policy&)
1064 {
1065    typedef typename tools::promote_args<T>::type result_type;
1066    typedef typename policies::evaluation<result_type, Policy>::type value_type;
1067    typedef typename policies::precision<result_type, Policy>::type precision_type;
1068    typedef typename policies::normalise<
1069       Policy,
1070       policies::promote_float<false>,
1071       policies::promote_double<false>,
1072       policies::discrete_quantile<>,
1073       policies::assert_undefined<> >::type forwarding_policy;
1074    typedef std::integral_constant<int,
1075       precision_type::value <= 0 ? 0 :
1076       precision_type::value <= 53 ? 53 :
1077       precision_type::value <= 64 ? 64 :
1078       precision_type::value <= 113 ? 113 : 0
1079    > tag_type;
1080 
1081    detail::zeta_initializer<value_type, forwarding_policy, tag_type>::force_instantiate();
1082 
1083    return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::zeta_imp(
1084       static_cast<value_type>(s),
1085       static_cast<value_type>(1 - static_cast<value_type>(s)),
1086       forwarding_policy(),
1087       tag_type()), "boost::math::zeta<%1%>(%1%)");
1088 }
1089 
1090 template <class T>
zeta(T s)1091 inline typename tools::promote_args<T>::type zeta(T s)
1092 {
1093    return zeta(s, policies::policy<>());
1094 }
1095 
1096 }} // namespaces
1097 
1098 #endif // BOOST_MATH_ZETA_HPP
1099 
1100 
1101 
1102