26 Numerics library [numerics]

26.9 Mathematical functions for floating-point types [c.math]

26.9.5 Mathematical special functions [sf.cmath]

26.9.5.18 Riemann zeta function [sf.cmath.riemann_zeta]

double riemann_zeta(double x); float riemann_zetaf(float x); long double riemann_zetal(long double x);

1

Effects: These functions compute the Riemann zeta function of their respective arguments x.

2

Returns: \[% \mathsf{\zeta}(x) = \left\{ \begin{array}{cl} \displaystyle \sum_{k=1}^\infty k^{-x}, & \mbox{for $x > 1$} \\ \\ \displaystyle \frac{1}{1 - 2^{1-x}} \sum_{k=1}^\infty (-1)^{k-1} k^{-x}, & \mbox{for $0 \le x \le 1$} \\ \\ \displaystyle 2^x \pi^{x-1} \sin(\frac{\pi x}{2}) \, \Gamma(1-x) \, \zeta(1-x), & \mbox{for $x < 0$} \end{array} \right. \; \] where x is x.