1[section:zeta Riemann Zeta Function] 2 3[h4 Synopsis] 4 5`` 6#include <boost/math/special_functions/zeta.hpp> 7`` 8 9 namespace boost{ namespace math{ 10 11 template <class T> 12 ``__sf_result`` zeta(T z); 13 14 template <class T, class ``__Policy``> 15 ``__sf_result`` zeta(T z, const ``__Policy``&); 16 17 }} // namespaces 18 19The return type of these functions is computed using the __arg_promotion_rules: 20the return type is `double` if T is an integer type, and T otherwise. 21 22[optional_policy] 23 24[h4 Description] 25 26 template <class T> 27 ``__sf_result`` zeta(T z); 28 29 template <class T, class ``__Policy``> 30 ``__sf_result`` zeta(T z, const ``__Policy``&); 31 32Returns the [@http://mathworld.wolfram.com/RiemannZetaFunction.html zeta function] 33of z: 34 35[equation zeta1] 36 37[graph zeta1] 38 39[graph zeta2] 40 41[h4 Accuracy] 42 43The following table shows the peak errors (in units of epsilon) 44found on various platforms with various floating point types, 45along with comparisons to the __gsl and __cephes libraries. 46Unless otherwise specified any floating point type that is narrower 47than the one shown will have __zero_error. 48 49[table_zeta] 50 51The following error plot are based on an exhaustive search of the functions domain, MSVC-15.5 at `double` precision, 52and GCC-7.1/Ubuntu for `long double` and `__float128`. 53 54[graph zeta__double] 55 56[graph zeta__80_bit_long_double] 57 58[graph zeta____float128] 59 60[h4 Testing] 61 62The tests for these functions come in two parts: 63basic sanity checks use spot values calculated using 64[@http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Zeta Mathworld's online evaluator], 65while accuracy checks use high-precision test values calculated at 1000-bit precision with 66[@http://shoup.net/ntl/doc/RR.txt NTL::RR] and this implementation. 67Note that the generic and type-specific 68versions of these functions use differing implementations internally, so this 69gives us reasonably independent test data. Using our test data to test other 70"known good" implementations also provides an additional sanity check. 71 72[h4 Implementation] 73 74All versions of these functions first use the usual reflection formulas 75to make their arguments positive: 76 77[equation zeta3] 78 79The generic versions of these functions are implemented using the series: 80 81[equation zeta6] 82 83When the significand (mantissa) size is recognised 84(currently for 53, 64 and 113-bit reals, plus single-precision 24-bit handled via promotion to double) 85then a series of rational approximations [jm_rationals] are used. 86 87For 0 < z < 1 the approximating form is: 88 89[equation zeta4] 90 91For a rational approximation /R(1-z)/ and a constant /C/: 92 93For 1 < z < 4 the approximating form is: 94 95[equation zeta5] 96 97For a rational approximation /R(n-z)/ and a constant /C/ and integer /n/: 98 99For z > 4 the approximating form is: 100 101[expression [zeta](z) = 1 + e[super R(z - n)]] 102 103For a rational approximation /R(z-n)/ and integer /n/, note that the accuracy 104required for /R(z-n)/ is not full machine-precision, but an absolute error 105of: /[epsilon]/R(0)/. This saves us quite a few digits when dealing with large 106/z/, especially when [epsilon] is small. 107 108Finally, there are some special cases for integer arguments, there are 109closed forms for negative or even integers: 110 111[equation zeta7] 112 113[equation zeta8] 114 115[equation zeta9] 116 117and for positive odd integers we simply cache pre-computed values as these are of great 118benefit to some infinite series calculations. 119 120[endsect] [/section:zeta Riemann Zeta Function] 121 122[/ :error_function The Error Functions] 123 124[/ 125 Copyright 2006 John Maddock and Paul A. Bristow. 126 Distributed under the Boost Software License, Version 1.0. 127 (See accompanying file LICENSE_1_0.txt or copy at 128 http://www.boost.org/LICENSE_1_0.txt). 129] 130