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

From cppreference.com
C++
Numerics library
Common mathematical functions
Mathematical special functions (C++17)
Mathematical constants (C++20)
Basic linear algebra algorithms (C++26)
Data-parallel types ( simd ) (C++26)
Floating-point environment (C++11)
Complex numbers
Numeric array ( valarray )
Pseudo-random number generation
Factor operations
(C++17)
(C++17)
Interpolations
(C++20)
(C++20)
Saturation arithmetic
(C++26)
(C++26)
(C++26)
(C++26)
(C++26)

Generic numeric operations
Bit operations
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
(C++20)
Mathematical 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)
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=1 n -num
  • For 0≤num≤1 ,
    1
    2 1-num -1
    Σ ∞n=1 (-1) n n -num
  • For num<0 , 2 num π num-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.
Retrieved from " https://en.cppreference.com/mwiki/index.php?title=cpp/numeric/special_functions/riemann_zeta&oldid=158821 "