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

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< cpp‎ | numeric‎ | special math

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

Run this code

    #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.

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