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

From cppreference.com
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 );
(1)
double      riemann_zeta( Integral arg );
(2)
1) Computes the Riemann zeta function of arg.
4) 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.

Contents

[edit] Parameters

arg - value of a floating-point or integral type

[edit] 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=1
    n-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)

[edit] 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

[edit] 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

[edit] Example

(works as shown with gcc 6.0)

#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)

[edit] External links

Weisstein, Eric W. "Riemann Zeta Function." From MathWorld--A Wolfram Web Resource.