std::riemann_zeta, std::riemann_zetaf, std::riemann_zetal
Defined in header <cmath>
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(1) | ||
float riemann_zeta ( float num ); double riemann_zeta ( double num ); |
(since C++17) (until C++23) |
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/* 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) |
Defined in header <cmath>
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template< class Integer > double riemann_zeta ( Integer num ); |
(A) | (since C++17) |
std::riemann_zeta
for all cv-unqualified floating-point types as the type of the parameter num.(since C++23)Contents |
[edit] Parameters
num | - | floating-point or value |
[edit] 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,
Σ∞1 21-num
-1
n=1 (-1)n
n-num
- For num<0, 2num
πnum-1
sin(
)Γ(1−num)ζ(1−num)πnum 2
[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 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)).
[edit] Example
#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
[edit] External links
Weisstein, Eric W. "Riemann Zeta Function." From MathWorld — A Wolfram Web Resource. |