NAME¶

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

Synopsis¶

double riemann_zeta( double arg );

double riemann_zeta( float arg );
double riemann_zeta( long double arg ); (1)
float riemann_zetaf( float arg );

long double riemann_zetal( long double arg );
double riemann_zeta( IntegralType arg ); (2)

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.

As all special functions, riemann_zeta is only guaranteed to be available in <cmath>
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.

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
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 TR 29124 but support TR 19768, provide this
function in the header tr1/cmath and namespace std::tr1.

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

Example¶

(works as shown with gcc 6.0)

// Run this code

#define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1
#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";
}

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.