std::riemann_zeta, std::riemann_zetaf, std::riemann_zetal

 

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Mathematical special functions

assoc_laguerreassoc_laguerrefassoc_laguerrel   assoc_legendreassoc_legendrefassoc_legendrel betabetafbetal comp_ellint_1comp_ellint_1fcomp_ellint_1l comp_ellint_2comp_ellint_2fcomp_ellint_2l comp_ellint_3comp_ellint_3fcomp_ellint_3l cyl_bessel_icyl_bessel_ifcyl_bessel_il

cyl_bessel_jcyl_bessel_jfcyl_bessel_jl   cyl_bessel_kcyl_bessel_kfcyl_bessel_kl cyl_neumanncyl_neumannfcyl_neumannl ellint_1ellint_1fellint_1l ellint_2ellint_2fellint_2l ellint_3ellint_3fellint_3l expintexpintfexpintl

hermitehermitefhermitel laguerrelaguerreflaguerrel legendrelegendreflegendrel riemann_zetariemann_zetafriemann_zetal sph_besselsph_besselfsph_bessell sph_legendresph_legendrefsph_legendrel sph_neumannsph_neumannfsph_neumannl

 

Defined in header <cmath>
	(1)
float       riemann_zeta ( float num ); double      riemann_zeta ( double num ); long double riemann_zeta ( long double num );			(since C++17) (until C++23)
/* floating-point-type */ riemann_zeta( /* floating-point-type */ num );			(since C++23)
float       riemann_zetaf( float num );		(2)	(since C++17)
long double riemann_zetal( long double num );		(3)	(since C++17)
Additional overloads
Defined in header <cmath>
template< class Integer > double      riemann_zeta ( Integer num );		(A)	(since C++17)

1-3) Computes the Riemann zeta function of num. The library provides overloads of std::riemann_zeta for all cv-unqualified floating-point types as the type of the parameter num. (since C++23)

A) Additional overloads are provided for all integer types, which are treated as double.

Parameters

num

-

floating-point or value

Return value

If no errors occur, value of the Riemann zeta function of num, ζ(num), defined for the entire real axis:

• For num>1, Σ∞
  n=1n-num
  
• For 0≤num≤1,

  1

  21-num -1

  Σ∞
  n=1 (-1)n
  n-num
  
• For num<0, 2num
  πnum-1
  sin(

  πnum

  2

  )Γ(1−num)ζ(1−num)

Error handling

Errors may be reported as specified in math_errhandling.

• If the argument is NaN, NaN is returned and domain error is not reported

Notes

Implementations that do not support C++17, but support ISO 29124:2010, provide this function if __STDCPP_MATH_SPEC_FUNCS__ is defined by the implementation to a value at least 201003L and if the user defines __STDCPP_WANT_MATH_SPEC_FUNCS__ before including any standard library headers.

Implementations that do not support ISO 29124:2010 but support TR 19768:2007 (TR1), provide this function in the header tr1/cmath and namespace std::tr1.

An implementation of this function is also available in boost.math.

The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their argument num of integer type, std::riemann_zeta(num) has the same effect as std::riemann_zeta(static_cast<double>(num)).

Example

Run this code

#include <cmath>
#include <iostream>
#include <numbers>
 
constexpr auto π = std::numbers::pi;
 
int main()
{
    // spot checks for well-known values
    std::cout << "ζ(-1)  = " << std::riemann_zeta(-1) << '\n'
              << "ζ(0)   = " << std::riemann_zeta(0) << '\n'
              << "ζ(1)   = " << std::riemann_zeta(1) << '\n'
              << "ζ(0.5) = " << std::riemann_zeta(0.5) << '\n'
              << "ζ(2)   = " << std::riemann_zeta(2) << '\n'
              << "π²/6   = " << π * π / 6 << '\n';
}

Output:

ζ(-1)  = -0.0833333
ζ(0)   = -0.5
ζ(1)   = inf
ζ(0.5) = -1.46035
ζ(2)   = 1.64493
π²/6   = 1.64493

External links

Weisstein, Eric W. "Riemann Zeta Function." From MathWorld — A Wolfram Web Resource.