NAME¶

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

Synopsis¶

Defined in header <cmath>
double riemann_zeta( double arg );

float riemann_zeta( float arg );
long double riemann_zeta( long double arg ); (1) (since C++17)
float riemann_zetaf( float arg );

long double riemann_zetal( long double arg );
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.