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

float       riemann_zeta ( float num );
double      riemann_zeta ( double num );
long double riemann_zeta ( long double num );

/* floating-point-type */ riemann_zeta( /* floating-point-type */ num );

float       riemann_zetaf( float num );

long double riemann_zetal( long double num );

template< class Integer >
double      riemann_zeta ( Integer num );

double

Parameters

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-1Σ∞
  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

#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';
}

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