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

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< cpp‎ | numeric‎ | special functions

Defined in header `<cmath>` |  |
| (1) |
float riemann_zeta ( float num ); double riemann_zeta ( double num ); long double riemann_zeta ( long double num ); |  | (since C++17) (until C++23)
/* floating-point-type */ riemann_zeta( /* floating-point-type */ num ); |  | (since C++23)
float riemann_zetaf( float num ); | (2) | (since C++17)
long double riemann_zetal( long double num ); | (3) | (since C++17)
Additional overloads |  |
Defined in header `<cmath>` |  |
template< class Integer > double riemann_zeta ( Integer num ); | (A) | (since C++17)

1-3) Computes the Riemann zeta function of num. The library provides overloads of `std::riemann_zeta` for all cv-unqualified floating-point types as the type of the parameter num.(since C++23)
A) Additional overloads are provided for all integer types, which are treated as double.

### Parameters

num | - | floating-point or value

### 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 -1

  Σ∞
  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

Run this code

    #include <cmath>
    #include <format>
    #include <iostream>
    #include <numbers>

    int main()
    {
        constexpr auto π = std::numbers::pi;

        // spot checks for well-known values
        for (const double x : {-1.0, 0.0, 1.0, 0.5, 2.0})
            std::cout << std::format("ζ({})\t= {:+.5f}\n", x, std::riemann_zeta(x));
        std::cout << std::format("π²/6\t= {:+.5f}\n", π * π / 6);
    }

Output:

    ζ(-1)   = -0.08333
    ζ(0)    = -0.50000
    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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