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  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.zetas.zeta"></a><a class="link" href="zeta.html" title="Riemann Zeta Function">Riemann Zeta Function</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.zetas.zeta.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.zetas.zeta.synopsis"></a></span><a class="link" href="zeta.html#math_toolkit.zetas.zeta.synopsis">Synopsis</a>
  32       </h5>
  33 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">zeta</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  34 </pre>
  35 <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
  36
  37 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  38 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
  39
  40 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  41 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  42
  43 <span class="special">}}</span> <span class="comment">// namespaces</span>
  44 </pre>
  45 <p>
  46         The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
  47         type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, and T otherwise.
  48       </p>
  49 <p>
  50         The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  51         be used to control the behaviour of the function: how it handles errors,
  52         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
  53         documentation for more details</a>.
  54       </p>
  55 <h5>
  56 <a name="math_toolkit.zetas.zeta.h1"></a>
  57         <span class="phrase"><a name="math_toolkit.zetas.zeta.description"></a></span><a class="link" href="zeta.html#math_toolkit.zetas.zeta.description">Description</a>
  58       </h5>
  59 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  60 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
  61
  62 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  63 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  64 </pre>
  65 <p>
  66         Returns the <a href="http://mathworld.wolfram.com/RiemannZetaFunction.html" target="_top">zeta
  67         function</a> of z:
  68       </p>
  69 <div class="blockquote"><blockquote class="blockquote"><p>
  70           <span class="inlinemediaobject"><img src="../../../equations/zeta1.svg"></span>
  71
  72         </p></blockquote></div>
  73 <div class="blockquote"><blockquote class="blockquote"><p>
  74           <span class="inlinemediaobject"><img src="../../../graphs/zeta1.svg" align="middle"></span>
  75
  76         </p></blockquote></div>
  77 <div class="blockquote"><blockquote class="blockquote"><p>
  78           <span class="inlinemediaobject"><img src="../../../graphs/zeta2.svg" align="middle"></span>
  79
  80         </p></blockquote></div>
  81 <h5>
  82 <a name="math_toolkit.zetas.zeta.h2"></a>
  83         <span class="phrase"><a name="math_toolkit.zetas.zeta.accuracy"></a></span><a class="link" href="zeta.html#math_toolkit.zetas.zeta.accuracy">Accuracy</a>
  84       </h5>
  85 <p>
  86         The following table shows the peak errors (in units of epsilon) found on
  87         various platforms with various floating point types, along with comparisons
  88         to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a> and
  89         <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries. Unless
  90         otherwise specified any floating point type that is narrower than the one
  91         shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
  92         zero error</a>.
  93       </p>
  94 <div class="table">
  95 <a name="math_toolkit.zetas.zeta.table_zeta"></a><p class="title"><b>Table&#160;8.76.&#160;Error rates for zeta</b></p>
  96 <div class="table-contents"><table class="table" summary="Error rates for zeta">
  97 <colgroup>
  98 <col>
  99 <col>
 100 <col>
 101 <col>
 102 <col>
 103 </colgroup>
 104 <thead><tr>
 105 <th>
 106               </th>
 107 <th>
 108                 <p>
 109                   GNU C++ version 7.1.0<br> linux<br> long double
 110                 </p>
 111               </th>
 112 <th>
 113                 <p>
 114                   GNU C++ version 7.1.0<br> linux<br> double
 115                 </p>
 116               </th>
 117 <th>
 118                 <p>
 119                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 120                 </p>
 121               </th>
 122 <th>
 123                 <p>
 124                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 125                 </p>
 126               </th>
 127 </tr></thead>
 128 <tbody>
 129 <tr>
 130 <td>
 131                 <p>
 132                   Zeta: Random values greater than 1
 133                 </p>
 134               </td>
 135 <td>
 136                 <p>
 137                   <span class="blue">Max = 0.846&#949; (Mean = 0.0833&#949;)</span><br>
 138                   <br> (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 5.45&#949; (Mean = 1&#949;))
 139                 </p>
 140               </td>
 141 <td>
 142                 <p>
 143                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 144                   2.1:</em></span> Max = 8.69&#949; (Mean = 1.03&#949;))
 145                 </p>
 146               </td>
 147 <td>
 148                 <p>
 149                   <span class="blue">Max = 0.846&#949; (Mean = 0.0833&#949;)</span>
 150                 </p>
 151               </td>
 152 <td>
 153                 <p>
 154                   <span class="blue">Max = 0.836&#949; (Mean = 0.093&#949;)</span>
 155                 </p>
 156               </td>
 157 </tr>
 158 <tr>
 159 <td>
 160                 <p>
 161                   Zeta: Random values less than 1
 162                 </p>
 163               </td>
 164 <td>
 165                 <p>
 166                   <span class="blue">Max = 7.03&#949; (Mean = 2.93&#949;)</span><br> <br>
 167                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 538&#949; (Mean = 59.3&#949;))
 168                 </p>
 169               </td>
 170 <td>
 171                 <p>
 172                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 173                   2.1:</em></span> Max = 137&#949; (Mean = 13.8&#949;))
 174                 </p>
 175               </td>
 176 <td>
 177                 <p>
 178                   <span class="blue">Max = 70.1&#949; (Mean = 17.1&#949;)</span>
 179                 </p>
 180               </td>
 181 <td>
 182                 <p>
 183                   <span class="blue">Max = 6.84&#949; (Mean = 3.12&#949;)</span>
 184                 </p>
 185               </td>
 186 </tr>
 187 <tr>
 188 <td>
 189                 <p>
 190                   Zeta: Values close to and greater than 1
 191                 </p>
 192               </td>
 193 <td>
 194                 <p>
 195                   <span class="blue">Max = 0.995&#949; (Mean = 0.5&#949;)</span><br> <br>
 196                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.9e+06&#949; (Mean = 5.11e+05&#949;))
 197                 </p>
 198               </td>
 199 <td>
 200                 <p>
 201                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 202                   2.1:</em></span> Max = 7.73&#949; (Mean = 4.07&#949;))
 203                 </p>
 204               </td>
 205 <td>
 206                 <p>
 207                   <span class="blue">Max = 0.995&#949; (Mean = 0.5&#949;)</span>
 208                 </p>
 209               </td>
 210 <td>
 211                 <p>
 212                   <span class="blue">Max = 0.994&#949; (Mean = 0.421&#949;)</span>
 213                 </p>
 214               </td>
 215 </tr>
 216 <tr>
 217 <td>
 218                 <p>
 219                   Zeta: Values close to and less than 1
 220                 </p>
 221               </td>
 222 <td>
 223                 <p>
 224                   <span class="blue">Max = 0.998&#949; (Mean = 0.508&#949;)</span><br> <br>
 225                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 8.53e+06&#949; (Mean = 1.87e+06&#949;))
 226                 </p>
 227               </td>
 228 <td>
 229                 <p>
 230                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 231                   2.1:</em></span> Max = 0.991&#949; (Mean = 0.28&#949;))
 232                 </p>
 233               </td>
 234 <td>
 235                 <p>
 236                   <span class="blue">Max = 0.998&#949; (Mean = 0.508&#949;)</span>
 237                 </p>
 238               </td>
 239 <td>
 240                 <p>
 241                   <span class="blue">Max = 0.991&#949; (Mean = 0.375&#949;)</span>
 242                 </p>
 243               </td>
 244 </tr>
 245 <tr>
 246 <td>
 247                 <p>
 248                   Zeta: Integer arguments
 249                 </p>
 250               </td>
 251 <td>
 252                 <p>
 253                   <span class="blue">Max = 9&#949; (Mean = 3.06&#949;)</span><br> <br>
 254                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 70.3&#949; (Mean = 17.4&#949;))
 255                 </p>
 256               </td>
 257 <td>
 258                 <p>
 259                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 260                   2.1:</em></span> Max = 3.75&#949; (Mean = 1.1&#949;))
 261                 </p>
 262               </td>
 263 <td>
 264                 <p>
 265                   <span class="blue">Max = 28&#949; (Mean = 5.62&#949;)</span>
 266                 </p>
 267               </td>
 268 <td>
 269                 <p>
 270                   <span class="blue">Max = 9&#949; (Mean = 3&#949;)</span>
 271                 </p>
 272               </td>
 273 </tr>
 274 </tbody>
 275 </table></div>
 276 </div>
 277 <br class="table-break"><p>
 278         The following error plot are based on an exhaustive search of the functions
 279         domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
 280         precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
 281         <span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
 282       </p>
 283 <div class="blockquote"><blockquote class="blockquote"><p>
 284           <span class="inlinemediaobject"><img src="../../../graphs/zeta__double.svg" align="middle"></span>
 285
 286         </p></blockquote></div>
 287 <div class="blockquote"><blockquote class="blockquote"><p>
 288           <span class="inlinemediaobject"><img src="../../../graphs/zeta__80_bit_long_double.svg" align="middle"></span>
 289
 290         </p></blockquote></div>
 291 <div class="blockquote"><blockquote class="blockquote"><p>
 292           <span class="inlinemediaobject"><img src="../../../graphs/zeta____float128.svg" align="middle"></span>
 293
 294         </p></blockquote></div>
 295 <h5>
 296 <a name="math_toolkit.zetas.zeta.h3"></a>
 297         <span class="phrase"><a name="math_toolkit.zetas.zeta.testing"></a></span><a class="link" href="zeta.html#math_toolkit.zetas.zeta.testing">Testing</a>
 298       </h5>
 299 <p>
 300         The tests for these functions come in two parts: basic sanity checks use
 301         spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Zeta" target="_top">Mathworld's
 302         online evaluator</a>, while accuracy checks use high-precision test values
 303         calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
 304         and this implementation. Note that the generic and type-specific versions
 305         of these functions use differing implementations internally, so this gives
 306         us reasonably independent test data. Using our test data to test other "known
 307         good" implementations also provides an additional sanity check.
 308       </p>
 309 <h5>
 310 <a name="math_toolkit.zetas.zeta.h4"></a>
 311         <span class="phrase"><a name="math_toolkit.zetas.zeta.implementation"></a></span><a class="link" href="zeta.html#math_toolkit.zetas.zeta.implementation">Implementation</a>
 312       </h5>
 313 <p>
 314         All versions of these functions first use the usual reflection formulas to
 315         make their arguments positive:
 316       </p>
 317 <div class="blockquote"><blockquote class="blockquote"><p>
 318           <span class="inlinemediaobject"><img src="../../../equations/zeta3.svg"></span>
 319
 320         </p></blockquote></div>
 321 <p>
 322         The generic versions of these functions are implemented using the series:
 323       </p>
 324 <div class="blockquote"><blockquote class="blockquote"><p>
 325           <span class="inlinemediaobject"><img src="../../../equations/zeta6.svg"></span>
 326
 327         </p></blockquote></div>
 328 <p>
 329         When the significand (mantissa) size is recognised (currently for 53, 64
 330         and 113-bit reals, plus single-precision 24-bit handled via promotion to
 331         double) then a series of rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
 332         by JM</a> are used.
 333       </p>
 334 <p>
 335         For 0 &lt; z &lt; 1 the approximating form is:
 336       </p>
 337 <div class="blockquote"><blockquote class="blockquote"><p>
 338           <span class="inlinemediaobject"><img src="../../../equations/zeta4.svg"></span>
 339
 340         </p></blockquote></div>
 341 <p>
 342         For a rational approximation <span class="emphasis"><em>R(1-z)</em></span> and a constant
 343         <span class="emphasis"><em>C</em></span>:
 344       </p>
 345 <p>
 346         For 1 &lt; z &lt; 4 the approximating form is:
 347       </p>
 348 <div class="blockquote"><blockquote class="blockquote"><p>
 349           <span class="inlinemediaobject"><img src="../../../equations/zeta5.svg"></span>
 350
 351         </p></blockquote></div>
 352 <p>
 353         For a rational approximation <span class="emphasis"><em>R(n-z)</em></span> and a constant
 354         <span class="emphasis"><em>C</em></span> and integer <span class="emphasis"><em>n</em></span>:
 355       </p>
 356 <p>
 357         For z &gt; 4 the approximating form is:
 358       </p>
 359 <div class="blockquote"><blockquote class="blockquote"><p>
 360           <span class="serif_italic">&#950;(z) = 1 + e<sup>R(z - n)</sup></span>
 361         </p></blockquote></div>
 362 <p>
 363         For a rational approximation <span class="emphasis"><em>R(z-n)</em></span> and integer <span class="emphasis"><em>n</em></span>,
 364         note that the accuracy required for <span class="emphasis"><em>R(z-n)</em></span> is not full
 365         machine-precision, but an absolute error of: /&#949;<span class="emphasis"><em>R(0)</em></span>.
 366         This saves us quite a few digits when dealing with large <span class="emphasis"><em>z</em></span>,
 367         especially when &#949; is small.
 368       </p>
 369 <p>
 370         Finally, there are some special cases for integer arguments, there are closed
 371         forms for negative or even integers:
 372       </p>
 373 <div class="blockquote"><blockquote class="blockquote"><p>
 374           <span class="inlinemediaobject"><img src="../../../equations/zeta7.svg"></span>
 375
 376         </p></blockquote></div>
 377 <div class="blockquote"><blockquote class="blockquote"><p>
 378           <span class="inlinemediaobject"><img src="../../../equations/zeta8.svg"></span>
 379
 380         </p></blockquote></div>
 381 <div class="blockquote"><blockquote class="blockquote"><p>
 382           <span class="inlinemediaobject"><img src="../../../equations/zeta9.svg"></span>
 383
 384         </p></blockquote></div>
 385 <p>
 386         and for positive odd integers we simply cache pre-computed values as these
 387         are of great benefit to some infinite series calculations.
 388       </p>
 389 </div>
 390 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 391 <td align="left"></td>
 392 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 393       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 394       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 395       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 396       Daryle Walker and Xiaogang Zhang<p>
 397         Distributed under the Boost Software License, Version 1.0. (See accompanying
 398         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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