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

double      riemann_zeta( double arg );
double      riemann_zeta( float arg );
double      riemann_zeta( long double arg );
float       riemann_zetaf( float arg );
long double riemann_zetal( long double arg );

double      riemann_zeta( Integral arg );

Parameters

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

  1-21-arg

  Σ∞
  n=1(-1)n-1
  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

#include <cmath>
#include <iostream>
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 = " << std::pow(std::acos(-1),2)/6 << ")\n";
}

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.