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

 

C++

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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>
double      riemann_zeta( double arg ); float       riemann_zeta( float arg ); long double riemann_zeta( long double arg ); float       riemann_zetaf( float arg ); long double riemann_zetal( long double arg );	(1)	(since C++17)
double      riemann_zeta( IntegralType arg );	(2)	(since C++17)

1) Computes the Riemann zeta function of arg.

2) A set of overloads or a function template accepting an argument of any integral type. Equivalent to (1) after casting the argument to double.

Parameters

arg

-

value of a floating-point or integral type

Return value

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

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

  1

  21-arg -1

  Σ∞
  n=1 (-1)n
  n-arg
  
• For arg<0, 2arg
  πarg-1
  sin(

  πarg

  2

  )Γ(1−arg)ζ(1−arg)

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

Example

Run this code

#include <cmath>
#include <iostream>
#include <numbers>
const auto π² = std::pow(std::numbers::pi,2);
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) << ' '
              << "(π²/6 = " << π²/6 << ")\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.